Search: a100071 -id:a100071
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1, 3, 11, 37, 123, 401, 1293, 4131, 13107, 41353, 129873, 406319, 1267093, 3940431, 12224579, 37845117, 116944371, 360771417, 1111332129, 3418840431, 10504903809, 32242682787, 98863833159, 302863592073, 927025884477, 2835306153351, 8665554849903
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OFFSET
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1,2
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COMMENTS
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LINKS
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FORMULA
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Binomial transform of A100071 starting [1, 2, 6, 12, 30, ...].
Inverse binomial transform of A037965 starting [1, 4, 18, 80, 350, ...].
a(n) = (n-1)! * [x^(n-1)] exp(x)*((1 + 2*x)*BesselI(0, 2*x) + 2*x*BesselI(1, 2*x)) for n>0, a(0) = 0. - Peter Luschny, Aug 26 2012
D-finite with recurrence (n-1)*a(n) = 3*(n-1)*a(n-1) +(n+1)*a(n-2) -3*(n-3)*a(n-3). - R. J. Mathar, Nov 09 2021
G.f.: x*(1-x)/((1-3*x)*sqrt((1+x)*(1-3*x))). - G. C. Greubel, May 28 2024
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EXAMPLE
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a(3) = 11 = (1, 2, 1) dot (1, 2, 6) = (1 + 4 + 6), where A100071 = (1, 2, 6, 12, 30, ...).
a(3) = 11 = (1, -2, 1) dot (1, 4, 18) = (1 - 8 + 18). - Gary W. Adamson, Nov 10 2007
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MATHEMATICA
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a[n_]:= a[n]= Sum[(-1)^(n-k-1)*Binomial[n-1, k]*(k+1)*Binomial[2*k, k], {k, 0, n-1}];
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PROG
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(Magma)
A134757:= func< n | (&+[(-1)^(n-k-1)*(k+1)^2*Binomial(n-1, k)*Catalan(k) : k in [0..n-1]]) >;
(SageMath)
def A134757(n): return sum((-1)^(n-k-1)*(k+1)*binomial(n-1, k)*binomial( 2*k, k) for k in range(n))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A215619
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a(n) is the number of consecutive terms of A100071, beginning with index n, which are divisible by n.
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+20
1
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4, 1, 6, 1, 8, 1, 4, 1, 12, 5, 14, 1, 4, 1, 18, 1, 20, 1, 4, 1, 24, 1, 6, 1, 4, 1, 30, 21, 32, 1, 12, 1, 8, 1, 38, 1, 14, 1, 42, 1, 44, 1, 6, 1, 48, 1, 8, 1, 4, 1, 54, 1, 6, 9, 4, 1, 60, 1, 62, 1, 4, 1, 6, 1, 68, 1, 4, 1, 72, 1, 74, 1, 4, 1, 12, 1, 80, 1, 4, 1
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OFFSET
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3,1
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COMMENTS
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a(n) = n+1 iff n is prime.
1 <= a(n) <= n+1.
{ n : a(2n)>1 } = { A058008 } \ { 1 }.
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LINKS
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MAPLE
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b:= proc(n) b(n):= n * binomial(n-1, floor((n-1)/2)) end:
a:= proc(n) local k;
for k from 0 while irem(b(n+k), n)=0 do od; k
end:
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MATHEMATICA
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b[n_] := n Binomial[n-1, Floor[(n-1)/2]];
a[n_] := Module[{k = 0}, While[Mod[b[n+k], n] == 0, k++]; k];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A056040
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Swinging factorial, a(n) = 2^(n-(n mod 2))*Product_{k=1..n} k^((-1)^(k+1)).
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+10
147
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1, 1, 2, 6, 6, 30, 20, 140, 70, 630, 252, 2772, 924, 12012, 3432, 51480, 12870, 218790, 48620, 923780, 184756, 3879876, 705432, 16224936, 2704156, 67603900, 10400600, 280816200, 40116600, 1163381400, 155117520, 4808643120, 601080390, 19835652870, 2333606220
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OFFSET
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0,3
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COMMENTS
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a(n) is the number of 'swinging orbitals' which are enumerated by the trinomial n over [floor(n/2), n mod 2, floor(n/2)].
Similar to but different from A001405(n) = binomial(n, floor(n/2)), a(n) = lcm(A001405(n-1), A001405(n)) (for n>0).
Exactly p consecutive multiples of p follow the least positive multiple of p if p is an odd prime. Compare with the similar property of A100071. - Peter Luschny, Aug 27 2012
a(n) is the number of vertices of the polytope resulting from the intersection of an n-hypercube with the hyperplane perpendicular to and bisecting one of its long diagonals. - Didier Guillet, Jun 11 2018 [Edited by Peter Munn, Dec 06 2022]
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LINKS
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FORMULA
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a(n) = n!/floor(n/2)!^2. [Essentially the original name.]
a(0) = 1, a(n) = n^(n mod 2)*(4/n)^(n+1 mod 2)*a(n-1) for n>=1.
O.g.f.: a(n) = SeriesCoeff_{n}((1+z/(1-4*z^2))/sqrt(1-4*z^2)).
P.g.f.: a(n) = PolyCoeff_{n}((1+z^2)^n+n*z*(1+z^2)^(n-1)).
a(2*n) = binomial(2*n,n); a(2*n+1) = (2*n+1)*binomial(2*n,n). Central terms of triangle A211226. - Peter Bala, Apr 10 2012
D-finite with recurrence: n*a(n) + (n-2)*a(n-1) + 4*(-2*n+3)*a(n-2) + 4*(-n+1)*a(n-3) + 16*(n-3)*a(n-4) = 0. - Alexander R. Povolotsky, Aug 17 2012
E.g.f.: U(0) where U(k)= 1 + x/(1 - x/(x + (k+1)*(k+1)/U(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Oct 19 2012
Central column of the coefficients of the swinging polynomials A162246. - Peter Luschny, Oct 22 2013
a(n) = hypergeometric([-n,-n-1,1/2],[-n-2,1],2)*2^(n-1)*(n+2). - Peter Luschny, Sep 22 2014
a(n) = 4^floor(n/2)*hypergeometric([-floor(n/2), (-1)^n/2], [1], 1). - Peter Luschny, May 19 2015
Sum_{n>=0} (-1)^n/a(n) = 4/3 - 4*Pi/(9*sqrt(3)). - Amiram Eldar, Mar 10 2022
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EXAMPLE
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a(10) = 10!/5!^2 = trinomial(10,[5,0,5]);
a(11) = 11!/5!^2 = trinomial(11,[5,1,5]).
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MAPLE
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SeriesCoeff := proc(s, n) series(s(w, n), w, n+2);
convert(%, polynom); coeff(%, w, n) end;
a1 := proc(n) local k;
2^(n-(n mod 2))*mul(k^((-1)^(k+1)), k=1..n) end:
a2 := proc(n) option remember;
`if`(n=0, 1, n^irem(n, 2)*(4/n)^irem(n+1, 2)*a2(n-1)) end;
a3 := n -> n!/iquo(n, 2)!^2;
g4 := z -> BesselI(0, 2*z)*(1+z);
a4 := n -> n!*SeriesCoeff(g4, n);
g5 := z -> (1+z/(1-4*z^2))/sqrt(1-4*z^2);
a5 := n -> SeriesCoeff(g5, n);
g6 := (z, n) -> (1+z^2)^n+n*z*(1+z^2)^(n-1);
a6 := n -> SeriesCoeff(g6, n);
a7 := n -> combinat[multinomial](n, floor(n/2), n mod 2, floor(n/2));
h := n -> binomial(n, floor(n/2)); # A001405
a8 := n -> ilcm(h(n-1), h(n));
F := [a1, a2, a3, a4, a5, a6, a7, a8];
for a in F do seq(a(i), i=0..32) od;
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MATHEMATICA
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f[n_] := 2^(n - Mod[n, 2])*Product[k^((-1)^(k + 1)), {k, n}]; Array[f, 33, 0] (* Robert G. Wilson v, Aug 02 2010 *)
f[n_] := If[OddQ@n, n*Binomial[n - 1, (n - 1)/2], Binomial[n, n/2]]; Array[f, 33, 0] (* Robert G. Wilson v, Aug 10 2010 *)
sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/f!]; (* or, twice faster: *) sf[n_] := n!/Quotient[n, 2]!^2; Table[sf[n], {n, 0, 32}] (* Jean-François Alcover, Jul 26 2013, updated Feb 11 2015 *)
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PROG
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(Magma) [(Factorial(n)/(Factorial(Floor(n/2)))^2): n in [0..40]]; // Vincenzo Librandi, Sep 11 2011
(Sage)
r, n = 1, 0
while True:
yield r
n += 1
r *= 4/n if is_even(n) else n
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A085750
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Determinant of the symmetric n X n matrix A defined by A[i,j] = |i-j| for 1 <= i,j <= n.
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+10
24
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0, -1, 4, -12, 32, -80, 192, -448, 1024, -2304, 5120, -11264, 24576, -53248, 114688, -245760, 524288, -1114112, 2359296, -4980736, 10485760, -22020096, 46137344, -96468992, 201326592, -419430400, 872415232, -1811939328, 3758096384, -7784628224, 16106127360
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OFFSET
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1,3
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COMMENTS
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The determinant of the distance matrix of a tree with vertex set {1,2,...,n}. The distance matrix is the n X n matrix in which the (i,j)-term is the number of edges in the unique path from vertex i to vertex j. [The matrix A in the definition is the distance matrix of the path-tree 1-2-...-n.]
Hankel transform of A100071. Also Hankel transform of C(2n-2,n-1)(-1)^(n-1). Inverse binomial transform of -n. - Paul Barry, Jan 11 2007
Pisano period lengths: 1, 1, 3, 1, 20, 3, 42, 1, 9, 20, 55, 3,156, 42, 60, 1,136, 9,171, 20, ... - R. J. Mathar, Aug 10 2012
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LINKS
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Emmanuel Briand, Luis Esquivias, Álvaro Gutiérrez, Adrián Lillo, and Mercedes Rosas, Determinant of the distance matrix of a tree, Proceedings of the 36th Conference on Formal Power Series and Algebraic Combinatorics (Bochum), Séminaire Lotharingien de Combinatoire 91B (2024) Article #29, 12 pp.
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FORMULA
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a(n) = (-1)^(n+1) * (n-1) * 2^(n-2) = (-1)^(n+1) * A001787(n-1).
a(n) = -4*a(n-1) - 4*a(n-2); a(1) = 0, a(1) = -1. - Philippe Deléham, Nov 03 2008
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MAPLE
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seq((-1)^(n-1)*(n-1)*2^(n-2), n = 1 .. 31);
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MATHEMATICA
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LinearRecurrence[{-4, -4}, {0, -1}, 40] (* Harvey P. Dale, Apr 14 2014 *)
CoefficientList[Series[-x/(1 + 2 x)^2, {x, 0, 40}], x] (* Vincenzo Librandi, Apr 15 2014 *)
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PROG
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CROSSREFS
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KEYWORD
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easy,sign
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AUTHOR
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Yuval Dekel (dekelyuval(AT)hotmail.com), Jul 21 2003
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EXTENSIONS
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STATUS
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approved
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A046212
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First numerator and then denominator of central elements of Leibniz's Harmonic Triangle.
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+10
20
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1, 1, 1, 6, 1, 30, 1, 140, 1, 630, 1, 2772, 1, 12012, 1, 51480, 1, 218790, 1, 923780, 1, 3879876, 1, 16224936, 1, 67603900, 1, 280816200, 1, 1163381400, 1, 4808643120, 1, 19835652870, 1, 81676217700, 1, 335780006100, 1, 1378465288200, 1
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OFFSET
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1,4
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 83, Problem 25.
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LINKS
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FORMULA
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A037965
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a(n) = n*binomial(2*n-2, n-1).
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+10
14
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0, 1, 4, 18, 80, 350, 1512, 6468, 27456, 115830, 486200, 2032316, 8465184, 35154028, 145608400, 601749000, 2481880320, 10218366630, 42004911960, 172427570700, 706905276000, 2894777105220, 11841673237680, 48394276165560, 197602337462400, 806190092077500
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OFFSET
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0,3
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COMMENTS
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REFERENCES
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The right-hand side of a binomial coefficient identity in H. W. Gould, Combinatorial Identities, Morgantown, 1972.
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LINKS
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FORMULA
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G.f.: Hypergeometric2F1([1/2, 2], [1], 4*x). - Paul Barry, Sep 03 2008
G.f.: x*(1-2*x)/(1-4*x)^(3/2);
a(n+1) = Sum_{k=0..n} binomial(2*k,k)*(4^(n-k) + 0^(n-k))/2. (End)
D-finite with recurrence (n-1)*a(n) - 2*(3*n-4)*a(n-1) + 4*(2*n-5)*a(n-2) = 0. - R. J. Mathar, Nov 30 2012
Sum_{n>=1} 1/a(n) = 4*Pi/(3*sqrt(3)) - Pi^2/9.
Sum_{n>=1} (-1)^(n+1)/a(n) = 8*log(phi)/sqrt(5) - 4*log(phi)^2, where phi is the golden ratio (A001622). (End)
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MATHEMATICA
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a[n_]:= n*Binomial[2*n-2, n-1]; Array[a, 30, 0] (* Amiram Eldar, Mar 10 2022 *)
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PROG
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(PARI) a(n) = n*binomial(2*n-2, n-1); \\ Joerg Arndt, Sep 04 2017
(Magma) [0] cat [n^2*Catalan(n-1): n in [1..30]]; // G. C. Greubel, Jun 19 2022
(SageMath) [n^2*catalan_number(n-1) for n in (0..30)] # G. C. Greubel, Jun 19 2022
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A331431
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Triangle read by rows: T(n,k) = (-1)^(n+k)*(n+k+1)*binomial(n,k)*binomial(n+k,k) for n >= k >= 0.
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+10
10
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1, -2, 6, 3, -24, 30, -4, 60, -180, 140, 5, -120, 630, -1120, 630, -6, 210, -1680, 5040, -6300, 2772, 7, -336, 3780, -16800, 34650, -33264, 12012, -8, 504, -7560, 46200, -138600, 216216, -168168, 51480, 9, -720, 13860, -110880, 450450, -1009008, 1261260, -823680, 218790
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OFFSET
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0,2
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COMMENTS
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Tables I, III, IV on pages 92 and 93 of Ser have integer entries and are A331430, A331431 (the present sequence), and A331432.
Given the system of equations 1 = Sum_{j=0..n} H(i, j) * x(j) for i = 2..n+2 where H(i,j) = 1/(i+j-1) for 1 <= i,j <= n is the n X n Hilbert matrix, then the solutions are x(j) = T(n, j). - Michael Somos, Mar 20 2020 [Corrected by Petros Hadjicostas, Jul 09 2020]
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REFERENCES
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J. Ser, Les Calculs Formels des Séries de Factorielles. Gauthier-Villars, Paris, 1933, p. 93. See Table III.
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LINKS
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FORMULA
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T(2*n, n) = (-1)^n*(3*n+1)!/(n!)^3 = (-1)^n*A331322(n).
Sum_{k=0..n} T(n, k) = A000290(n+1) (row sums).
Sum_{k=0..n}((-1)^k*T(n, k) = (-1)^n*A108666(n+1) (alternating row sums).
Sum_{k=0..n} T(n-k, k) = (-1)^n*A109188(n+1) (diagonal sums).
2^n*Sum_{k=0..n} T(n, k)/2^k = (-1)^floor(n/2)*A100071(n+1) (positive half sums).
(-2)^n*Sum_{k=0..n} T(n, k)/(-2)^k = A331323(n) (negative half sums).
T(n,k) = (-1)^(n+k)*(n+k+1)!/((k!)^2*(n-k)!), for n >= k >= 0. - N. J. A. Sloane, Jan 18 2020
Sum_{k=0..n} T(n,k)/(i+k) = 1 for i = 1..n+1.
These are special cases of the following formula that is alluded to (in some way) in Ser's book:
1 - Sum_{k=0..n} T(n,k)/(x + k) = (x-1)*...*(x-(n + 1))/(x*(x+1)*...*(x+n)).
Because T(n,k) = (-1)^(n+1)*(n + k + 1)*A331430(n,k) and Sum_{k=0..n} A331430(n,k) = (-1)^(n+1), one may derive this formula from Ser's second formula stated in A331430. (End)
T(2*n+1, n) = (-2)*(-27)^n*Pochhammer(4/3, n)*Pochhammer(5/3, n)/(n!*(n+1)!). - G. C. Greubel, Mar 22 2022
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EXAMPLE
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Triangle begins:
1;
-2, 6;
3, -24, 30;
-4, 60, -180, 140;
5, -120, 630, -1120, 630;
-6, 210, -1680, 5040, -6300, 2772;
7, -336, 3780, -16800, 34650, -33264, 12012;
-8, 504, -7560, 46200, -138600, 216216, -168168, 51480;
9, -720, 13860, -110880, 450450, -1009008, 1261260, -823680, 218790;
...
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MAPLE
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gf := k -> (1+x)^(-2*(k+1)): ser := k -> series(gf(k), x, 32):
T := (n, k) -> ((2*k+1)!/(k!)^2)*coeff(ser(k), x, n-k):
S:=(n, k)->(-1)^(n+k)*(n+k+1)!/((k!)^2*(n-k)!);
rho:=n->[seq(S(n, k), k=0..n)];
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MATHEMATICA
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Table[(-1)^(n+k)*(n+k+1)*Binomial[2*k, k]*Binomial[n+k, n-k], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 22 2022 *)
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PROG
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(Magma) [(-1)^(n+k)*(k+1)*(2*k+1)*Binomial(n+k+1, n-k)*Catalan(k): k in [0..n], n in [0..15]]; // G. C. Greubel, Mar 22 2022
(Sage) flatten([[(-1)^(n+k)*(2*k+1)*binomial(2*k, k)*binomial(n+k+1, n-k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Mar 22 2022
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A137762
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Central elements in writing first the numerator and then the denominator (left to right) of Leibniz's harmonic-like triangle.
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+10
7
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1, 1, 5, 6, 31, 30, 209, 140, 1471, 630, 10625, 2772
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OFFSET
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1,3
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LINKS
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FORMULA
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EXAMPLE
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1/1; --> 1 1
1/2, 1/2; -->
1/3, 5/6, 1/3; --> 5 6
1/4, 7/12, 7/12, 1/4; --> ...
1/5, 9/20, 31/30, 9/20, 1/5;
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CROSSREFS
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KEYWORD
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nonn,tabf,frac
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AUTHOR
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STATUS
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approved
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A137763
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Central elements in writing first the denominator and then the numerator(left to right) of Leibniz's harmonic-like triangle.
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+10
7
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1, 1, 6, 5, 30, 31, 140, 209, 630, 1471, 2772, 10625
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OFFSET
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1,3
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LINKS
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FORMULA
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EXAMPLE
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1/1; --> 1 1
1/2, 1/2; -->
1/3, 5/6, 1/3; --> 6 5
1/4, 7/12, 7/12, 1/4; --> ...
1/5, 9/20, 31/30, 9/20, 1/5;
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CROSSREFS
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Cf. A046201, A046204, A046205, A046206, A046208, A046212, A137752, A137753, A137754, A137755, A137756, A137757, A137758, A137759, A137760, A137761, A137762.
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KEYWORD
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nonn,tabf,frac,less
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AUTHOR
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STATUS
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approved
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A107230
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A number triangle of inverse Chebyshev transforms.
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+10
5
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1, 1, 1, 2, 2, 1, 3, 6, 3, 1, 6, 12, 12, 4, 1, 10, 30, 30, 20, 5, 1, 20, 60, 90, 60, 30, 6, 1, 35, 140, 210, 210, 105, 42, 7, 1, 70, 280, 560, 560, 420, 168, 56, 8, 1, 126, 630, 1260, 1680, 1260, 756, 252, 72, 9, 1, 252, 1260, 3150, 4200, 4200, 2520, 1260, 360, 90, 10, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,4
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COMMENTS
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First column is A001405, second column is A100071, third column is A107231. Row sums are A005773(n+1), diagonal sums are A026003. The inverse Chebyshev transform concerned takes a g.f. g(x)->(1/sqrt(1-4x^2))g(xc(x^2)) where c(x) is the g.f. of A000108. It transforms a(n) to b(n) = Sum_{k=0..floor(n/2)} binomial(n,k)*a(n-2k). Then a(n) = Sum_{k=0..floor(n/2)} (n/(n-k))*(-1)^k*binomial(n-k,k) *b(n-2k).
Triangle read by rows: T(n,k) is the number of paths of length n with steps U=(1,1), D=(1,-1) and H=(1,0), starting at (0,0), staying weakly above the x-axis (i.e., left factors of Motzkin paths) and having k H steps. Example: T(3,1)=6 because we have HUD. HUU, UDH, UHD, UHU and UUH. Sum_{k=0..n} k*T(n,k) = A132894(n). - Emeric Deutsch, Oct 07 2007
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LINKS
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FORMULA
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T(n,k) = binomial(n,k)*binomial(n-k, floor((n-k)/2)).
G.f.: G=G(t,z) satisfies z*(1-2*z-t*z)*G^2+(1-2*z-t*z)*G-1=0. - Emeric Deutsch, Oct 07 2007
E.g.f.: exp(x*y)*(BesselI(0,2*x)+BesselI(1,2*x)). - Vladeta Jovovic, Dec 02 2008
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EXAMPLE
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Triangle begins
1;
1, 1;
2, 2, 1;
3, 6, 3, 1;
6, 12, 12, 4, 1;
10, 30, 30, 20, 5, 1;
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MAPLE
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T:=proc(n, k) options operator, arrow: binomial(n, k)*binomial(n-k, floor((1/2)*n-(1/2)*k)) end proc: for n from 0 to 11 do seq(T(n, k), k=0..n) end do; # yields sequence in triangular form - Emeric Deutsch, Oct 07 2007
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MATHEMATICA
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Table[Binomial[n, k]*Binomial[n-k, Floor[(n-k)/2]], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 11 2019 *)
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PROG
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(PARI) T(n, k) = binomial(n, k)*binomial(n-k, (n-k)\2); \\ Michel Marcus, Feb 10 2019
(Magma) [[Binomial(n, k)*Binomial(n-k, Floor((n-k)/2)): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Feb 11 2019
(Sage) [[binomial(n, k)*binomial(n-k, floor((n-k)/2)) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Feb 11 2019
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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