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Search: a046714 -id:a046714
Displaying 1-4 of 4 results found. page 1
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A046748 Row sums of triangle A046521. +10
7
1, 3, 13, 61, 295, 1447, 7151, 35491, 176597, 880125, 4390901, 21920913, 109486993, 547018941, 2733608905, 13662695645, 68294088535, 341399727335, 1706739347095, 8532741458075, 42660172763995, 213287735579135, 1066389745361635, 5331765761680895 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Hankel transform is A082761. - Paul Barry, Apr 14 2010
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
T.-X. He, L. W. Shapiro, Fuss-Catalan matrices, their weighted sums, and stabilizer subgroups of the Riordan group, Lin. Alg. Applic. 532 (2017) 25-41, p 35.
FORMULA
a(n) = binomial(2*n, n)*Sum_{k=0..n} binomial(n, k)/binomial(2*k, k).
a(n) = 5^n - 2*A046714(n-1), A046714(-1) := 0.
a(n) = 5*a(n-1) - 2*A000108(n-1).
G.f.: sqrt(1-4*x)/(1-5*x).
a(n) = (3*(3*n-2)/n)*a(n-1) - (10*(2*n-3)/n)*a(n-2), n >= 1, a(-1) := 0, a(0)=1 (homogeneous recursion).
a(n) = binomial(2*n,n)*hypergeom([ -n,1 ],[ 1/2 ],-1/4) (hypergeometric 2F1 form).
0 = a(n)*(+400*a(n+1) - 330*a(n+2) + 50*a(n+3)) + a(n+1)*(-30*a(n+1) + 71*a(n+2) - 15*a(n+3)) + a(n+2)*(-3*a(n+2) + a(n+3)) for all n in Z. - Michael Somos, May 25 2014
a(n) ~ 5^(n - 1/2). - Vaclav Kotesovec, Jul 07 2016
D-finite with recurrence n*a(n) +3*(-3*n+2)*a(n-1) +10*(2*n-3)*a(n-2)=0. - R. J. Mathar, Jul 23 2017
EXAMPLE
G.f. = 1 + 3*x + 13*x^2 + 61*x^3 + 295*x^4 + 1447*x^5 + 7151*x^6 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ Sqrt[ 1 - 4 x] / (1 - 5 x), {x, 0, n}]; (* Michael Somos, May 25 2014 *)
a[ n_] := Binomial[ 2 n, n] Hypergeometric2F1[ -n, 1, 1/2, -1/4]; (* Michael Somos, May 25 2014 *)
PROG
(PARI) {a(n) = if( n<0, 0, polcoeff( sqrt( 1 - 4*x + x * O(x^n)) / (1 - 5*x), n))}; /* Michael Somos, May 25 2014 */
(Magma)
R<x>:=PowerSeriesRing(Rationals(), 40);
Coefficients(R!( Sqrt(1-4*x)/(1-5*x) )); // G. C. Greubel, Jul 28 2024
(SageMath)
def A046748_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( sqrt(1-4*x)/(1-5*x) ).list()
A046748_list(40) # G. C. Greubel, Jul 28 2024
CROSSREFS
Cf. A000108, A046521, A046714, A082761.
KEYWORD
easy,nonn
AUTHOR
Wolfdieter Lang, Dec 11 1999
STATUS
approved
A046814 Row sums of triangle A046527. +10
2
1, 2, 8, 37, 179, 881, 4369, 21746, 108444, 541362, 2704158, 13512392, 67534828, 337584992, 1687627800, 8437136085, 42182258715, 210899507685, 1054456597965, 5272139698215, 26360193558735, 131799177579015 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: c(x) * (1-4*x) / (1-5*x), where c(x) = g.f. for Catalan A000108.
a(n) = C(n) + A046714(n-1) with A046714(-1) = 0 and C(n) = A000108(n) are the Catalan numbers.
a(n) = C(n) + (5^n - A046748(n))/2.
a(n) = 5*a(n-1) - 3*C(n)/(2*n-1), a(0)=1.
D-finite with recurrence a(n) = (9*n-1)*a(n-1)/(n+1) - 10*(2*n-3)*a(n-2)/(n+1), n >= 2, a(0)=1, a(1)=2.
MATHEMATICA
CoefficientList[Series[(1-4*x)*(1-Sqrt[1-4*x])/(2*x*(1-5*x)), {x, 0, 40}], x] (* G. C. Greubel, Jul 28 2024 *)
PROG
(Magma)
[n le 1 select 1 else 5*Self(n-1) - 3*Catalan(n-1)/(2*n-3): n in [1..40]]; // G. C. Greubel, Jul 28 2024
(SageMath)
@CachedFunction
def A046814(n): return 1 if n==0 else 5*A046814(n-1) - 3*catalan_number(n)/(2*n-1)
[A046814(n) for n in range(41)] # G. C. Greubel, Jul 28 2024
CROSSREFS
Cf. A000108, A046527, A046714, A046748.
KEYWORD
easy,nonn
AUTHOR
Wolfdieter Lang
EXTENSIONS
Offset corrected by Sean A. Irvine, Apr 25 2021
STATUS
approved
A046885 Row sums of triangle A046658. +10
2
1, 4, 18, 85, 411, 2013, 9933, 49236, 244750, 1218888, 6077644, 30329434, 151439158, 756452890, 3779590010, 18888255205, 94405918355, 471899946985, 2359022096225, 11793343217935, 58960151969255, 294776293579255 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
FORMULA
a(n) = 2*5^(n-1) - A046714(n-1) = (A046748(n) - 5^(n-1))/2.
G.f.: x*(2 - c(x))/(1-5*x), where c(x) is the g.f. of A000108 (Catalan numbers).
Inhomogeneous recursion: a(n) = 5*a(n-1) - C(n-1), n >= 2, a(1)=1; C(n) = A000108(n) (Catalan).
Homogeneous recursion: a(n) = (3*(3*n-2)/n)*a(n-1) - (10*(2*n-3)/n)*a(n-2), n >= 3, a(1)=1, a(2)=4.
MATHEMATICA
Rest@CoefficientList[Series[Sqrt[1-4*x]*(1-Sqrt[1-4*x])/(2*(1-5*x)), {x, 0, 40}], x] (* G. C. Greubel, Jul 28 2024 *)
PROG
(Magma)
[n le 1 select 1 else 5*Self(n-1) - Catalan(n-1): n in [1..40]]; // G. C. Greubel, Jul 28 2024
(SageMath)
@CachedFunction
def A046885(n): return 1 if n==1 else 5*A046885(n-1) - catalan_number(n-1)
[A046885(n) for n in range(1, 41)] # G. C. Greubel, Jul 28 2024
CROSSREFS
Cf. A000108, A046658, A046714, A046748.
KEYWORD
easy,nonn
AUTHOR
Wolfdieter Lang
STATUS
approved
A271453 Triangle read by rows of coefficients of polynomials C_n(x) = Sum_{k=0..n} (2*k)!*(x - 1)^(n-k)/((k + 1)!*k!). +10
1
1, 0, 1, 2, -1, 1, 3, 3, -2, 1, 11, 0, 5, -3, 1, 31, 11, -5, 8, -4, 1, 101, 20, 16, -13, 12, -5, 1, 328, 81, 4, 29, -25, 17, -6, 1, 1102, 247, 77, -25, 54, -42, 23, -7, 1, 3760, 855, 170, 102, -79, 96, -65, 30, -8, 1, 13036, 2905, 685, 68, 181, -175, 161, -95, 38, -9, 1, 45750, 10131, 2220, 617, -113, 356, -336, 256, -133, 47, -10, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
The polynomials C_n(x) have generating function G(x,t) = (1 - sqrt(1 - 4*t))/(2*t*(1 + t - x*t)) = 1 + x*t + (x^2 - x + 2)*t^2 + (x^3 - 2*x^2 + 3*x + 3)*t^3 + ...
C_n(x) can be defined by the recurrence relation C_n(x) = (x - 1)*C_(n-1)(x) + (2n)!/((n + 1)!*n!), C_0(x) = 1 or the equivalent form C_n(x) = (x - 1)*C_(n-1)(x) + C_n(1), C_0(x) = 1.
C_n(x) can be defined as convolution of Catalan numbers and powers of (x - 1).
Discriminants of C_n(x) gives the sequence: 1, 1, -7, -543, 533489, 7080307052, -1318026434480736, -3526797951451513832247, 137992774365121594001729513153, ...
C_n(0) = A032357(n).
C_n(1) = C_n(x) - (x - 1)*C_(n-1)(x) = A000108(n).
C_n(2) = Sum_{m=0..n} C_1(m) = A014137(n).
C_n(3) = A014318(n).
C_n(5) = A000346(n).
C_n(6) = A046714(n).
LINKS
G. C. Greubel, Rows n=0..100 of triangle, flattened
Ilya Gutkovskiy, Polynomials C_n(x)
Eric Weisstein's World of Mathematics, Catalan Number
FORMULA
For triangle: T(n,n)=1, T(n,0) = Sum_{k=0..n} (-1)^(n-k)*(2*k)!/(k! * (k+1)!), T(n, k) = T(n-1, k-1) - T(n-1, k). - G. C. Greubel, Nov 04 2018
EXAMPLE
Triangle begins:
1;
0, 1;
2, -1, 1;
3, 3, -2, 1;
11, 0, 5, -3, 1;
31, 11, -5, 8, -4, 1;
...
The first few polynomials are:
C_0(x) = 1;
C_1(x) = x;
C_2(x) = x^2 - x + 2;
C_3(x) = x^3 - 2*x^2 + 3*x + 3;
C_4(x) = x^4 - 3*x^3 + 5*x^2 + 11;
C_5(x) = x^5 - 4*x^4 + 8*x^3 - 5*x^2 + 11*x + 31;
...
MATHEMATICA
CoefficientList[RecurrenceTable[{c[0] == 1, c[n] == (x - 1) c[n - 1] + CatalanNumber[n]}, c, {n, 11}], x]
T[n_, n_]:= 1; T[n_, 0]:= (-1)^n*Sum[CatalanNumber[k]*(-1)^k, {k, 0, n}]; T[n_, k_]:= T[n - 1, k - 1] - T[n - 1, k]; Table[T[n, k], {n, 0, 5}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 04 2018 *)
PROG
(PARI) {T(n, k) = if(k==n, 1, if(k==0, sum(j=0, n, (-1)^(n-j)*(2*j)!/(j!*(j+1)!)), T(n-1, k-1) - T(n-1, k))) };
for(n=0, 10, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Nov 04 2018
CROSSREFS
Cf. A000108, A130595.
KEYWORD
sign,tabl,easy
AUTHOR
Ilya Gutkovskiy, Apr 09 2016
STATUS
approved
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Last modified August 29 11:15 EDT 2024. Contains 375512 sequences. (Running on oeis4.)