Search: a026374 -id:a026374
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A026378
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a(n) = number of integer strings s(0),...,s(n) counted by array T in A026374 that have s(n)=1; also a(n) = T(2n-1,n-1).
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+20
28
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1, 4, 17, 75, 339, 1558, 7247, 34016, 160795, 764388, 3650571, 17501619, 84179877, 406020930, 1963073865, 9511333155, 46169418195, 224484046660, 1093097083475, 5329784874185, 26018549129545, 127154354598330, 622031993807565
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OFFSET
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1,2
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COMMENTS
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Number of lattice paths from (0,0) to the line x=n-1 that do not go below the line y=0 and consist of steps U=(1,1), D=(1,-1) and three types of steps H=(1,0) (left factors of 3-Motzkin steps). Example: a(3)=17 because we have UD, UU, 9 HH paths, 3 HU paths and 3 UH paths. - Emeric Deutsch, Jan 22 2004
Also a(n) = number of integer strings s(0), ..., s(n) counted by array U in A026386 that have s(n)=1; a(n) = U(2n-1, n-1).
The Hankel transform of [1,1,4,17,75,339,1558,...] is [1,3,8,21,55,144,377,...] (see A001906). - Philippe Deléham, Apr 13 2007
Number of peaks in all skew Dyck paths of semilength n. A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of the path is defined to be the number of its steps. Example: a(2)=4 because in the 3 (=A002212(2)) skew Dyck paths (UD)(UD), U(UD)D and U(UD)L we have altogether 4 peaks (shown between parentheses). - Emeric Deutsch, Jul 25 2007
Convolved with A007317, (1, 2, 5, 15, 51, ...) = A026376: (1, 6, 30, 144, ...)
Equals A026375, (1, 3, 11, 45, 195, ...) convolved with A002212 prefaced with
a 1: (1, 1, 3, 10, 36, 137, ...). (End)
The array belongs to an interpolated family of arrays associated to the Catalan A000108 (t=1), and Riordan, or Motzkin sums A005043 (t=0), with the interpolating o.g.f. [1-sqrt(1-4x/(1+(1-t)x))]/2 and inverse x(1-x)/[1+(t-1)x(1-x)]. See A091867 for more info on this family. Here the interpolation is t=-4 (mod signs in the results).
Let C(x) = [1 - sqrt(1-4x)]/2, an o.g.f. for the Catalan numbers A000108, with inverse Cinv(x) = x*(1-x) and P(x,t) = x/(1+t*x) with inverse P(x,-t).
O.g.f: G(x) = [-1 + sqrt(1 + 4*x/(1-5x))]/2 = -C[P(-x,5)].
Inverse O.g.f: Ginv(x) = x*(1+x)/[1 + 5x*(1+x)] = -P(Cinv(-x),-5) (signed A039717). (End)
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LINKS
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E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.
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FORMULA
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G.f.: (1/2)/(5*x^2-x)*(1-5*x-(1-6*x+5*x^2)^(1/2)). E.g.f.: exp(3*x)*(BesselI(0, 2*x)+BesselI(1, 2*x)). - Vladeta Jovovic, Oct 03 2003
G.f.: [(1-z)/sqrt(1-6z+5z^2)-1]/2 = z + 4z^2 + 17z^3 + ... - Emeric Deutsch, Jan 22 2004
a(n) = coefficient of t^n in (1+t)(1+3t+t^2)^(n-1). - Emeric Deutsch, Jan 30 2004
a(n) = [2(3n-2)a(n-1) - 5(n-2)a(n-2)]/n for n>=2; a(0)=0, a(1)=1. - Emeric Deutsch, Mar 18 2004
a(n+1) = sum(k=0, n, binomial(n, k)*sum(i=0, k, binomial(k+i, i))). - Benoit Cloitre, Aug 06 2004
a(n+1) = sum(k=0, n, binomial(n, k)*binomial(2*k+1, k+1)). - Benoit Cloitre, Aug 06 2004
G.f.: (1/(1-5x))*c(-x/(1-5x)), c(x) the g.f. of A000108;
a(n) = sum{k=0..n, C(n,k)*(-1)^k*A000108(k)*5^(n-k)} (offset 0). (End)
G.f. 1/(1 - 3x - x(1 - x)/(1 - x - x(1 - x)/(1 - x - x(1 - x)/(1 - x - x(1 - x)/(1...(continued fraction). - Aoife Hennessy (aoife.hennessy(AT)gmail.com), Jul 02 2010
a(n) = (-1)^n*(GegenbauerC(n-2,-n+1,3/2) - GegenbauerC(n-1,-n+1,3/2)). - Peter Luschny, May 13 2016
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MAPLE
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a := n -> (-1)^n*simplify(GegenbauerC(n-2, -n+1, 3/2) - GegenbauerC(n-1, -n+1, 3/2)): seq(a(n), n=1..23); # Peter Luschny, May 13 2016
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MATHEMATICA
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CoefficientList[Series[(1/2)/(5*x^2-x)*(1-5*x-(1-6*x+5*x^2)^(1/2)), {x, 0, 30}], x] (* Vincenzo Librandi, May 13 2012 *)
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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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A026376
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a(n) is the number of integer strings s(0),...,s(n) counted by array T in A026374 that have s(n)=2; also a(n) = T(2n,n-1).
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+20
12
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1, 6, 30, 144, 685, 3258, 15533, 74280, 356283, 1713690, 8263596, 39938616, 193419915, 938430990, 4560542550, 22195961280, 108171753355, 527816696850, 2578310320610, 12607504827600, 61706212037295, 302275142049870, 1481908332595625, 7270432009471224
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OFFSET
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1,2
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COMMENTS
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Number of Schroeder paths (i.e., consisting of steps U=(1,1), D=(1,-1) and H=(2,0) and never going below the x-axis) from (0,0) to (2n+2,0), with exactly one peak at an even level. E.g., a(2)=6 because we have UUDDH, HUUDD, UDUUDD, UUDDUD, UUDHD and UHUDD. - Emeric Deutsch, Dec 28 2003
Number of left steps in all skew Dyck paths of semilength n+1. A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of the path is defined to be the number of its steps. Example: a(2)=6 because in the 10 (=A002212(3)) skew Dyck paths of semilength 3 ( namely UDUUDL, UUUDLD, UUDUDL, UUUDDL, UUUDLL and five Dyck paths that have no left steps) we have altogether 6 left steps. - Emeric Deutsch, Aug 05 2007
Equals A026378 (1, 4, 17, 75, ...) convolved with A007317 (1, 2, 5, 15, 51, ...).
Equals A081671 (1, 3, 11, 45, ...) convolved with A002212 (1, 3, 10, 36, 137, ...).
(End)
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LINKS
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Emeric Deutsch, Emanuele Munarini, and Simone Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.
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FORMULA
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E.g.f.: exp(3x)*I_1(2x), where I_1 is Bessel function. - Michael Somos, Sep 09 2002
G.f.: (1 - 3*z - t)/(2*z*t) where t = sqrt(1-6*z+5*z^2). - Emeric Deutsch, May 25 2003
a(n) = [t^(n+1)](1+3t+t^2)^n. a := n -> Sum_{j=ceiling((n+1)/2)..n} 3^(2j-n-1)*binomial(n, j)*binomial(j, n+1-j). - Emeric Deutsch, Jan 30 2004
a(n) = Sum_{k=0..n} binomial(n, k)*binomial(2k, k+1). - Paul Barry, Sep 20 2004
D-finite with recurrence (n+1)*a(n) - 9*n*a(n-1) + (23*n-27)*a(n-2) + 15*(-n+2)*a(n-3) = 0. - R. J. Mathar, Dec 02 2012
a(n) = n*hypergeometric([1, 3/2, 1-n],[1, 3],-4). - Peter Luschny, Sep 16 2014
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MAPLE
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a := n -> simplify(GegenbauerC(n-1, -n, -3/2)):
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MATHEMATICA
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Rest[CoefficientList[Series[(1-3*x-Sqrt[1-6*x+5*x^2])/(2*x*Sqrt[1-6*x+5*x^2]), {x, 0, 20}], x]] (* Vaclav Kotesovec, Feb 13 2014 *)
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PROG
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(PARI) a(n)=if(n<0, 0, polcoeff((1+3*x+x^2)^n, n-1))
(Sage)
A026376 = lambda n : n*hypergeometric([1, 3/2, 1-n], [1, 3], -4)
(Sage) # Recurrence:
x, y, n = 1, 1, 1
while True:
x, y = y, ((6*n + 3)*y - (5*n - 5)*x) / (n + 2)
yield n*x
n += 1
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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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A026377
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a(n) = number of integer strings s(0),...,s(n) counted by array T in A026374 that have s(n)=4; also a(n) = T(2n,n-2).
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+20
3
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1, 9, 58, 330, 1770, 9198, 46928, 236736, 1185645, 5909805, 29362806, 145570230, 720606705, 3563543025, 17610412600, 86989143480, 429579843435, 2121099312195, 10472653252550, 51708363376950, 255326054688320
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OFFSET
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2,2
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COMMENTS
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Number of lattice paths from (0,0) to (2n,4), using steps U=(1,1), D=(1,-1) and at even levels(zero, positive and negative) also H=(2,0). Example: a(3)=9 because we have UUUUUD, UUUUDU, UUUDUU, UUDUUU, UDUUUU, DUUUUU, HUUUU, UUHUU and UUUUH. - Emeric Deutsch, Jan 30 2004
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LINKS
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FORMULA
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a(n) = [t^(n+2)](1+3t+t^2)^n.
a(n) = Sum_{j=ceiling((n+2)/2)..n} (3^(2j-n-2)*binomial(n, j)*binomial(j, n+2-j)). (End)
E.g.f.: exp(3x) * BesselI(2, 2x);
a(n) = Sum_{k=0..n} binomial(n, k)*binomial(2k, k+2). (End)
Conjecture: n*(n+4)*a(n) - 3*(n+2)*(2*n+3)*a(n-1) + 5*(n+2)*(n+1)*a(n-2) = 0. - R. J. Mathar, Nov 24 2012
G.f.: (3*x - 1 + (1-6*x+7*x^2)/sqrt(5*x^2-6*x+1))/(2*x^2). - Mark van Hoeij, Apr 18 2013
Assuming offset 0: a(n) = C(2*n+4,n)*hypergeom([-n,-n-4],[-3/2-n],-1/4). - Peter Luschny, May 09 2016
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MAPLE
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series( (3*x-1+(1-6*x+7*x^2)/sqrt(5*x^2-6*x+1))/(2*x^2), x=0, 30); # Mark van Hoeij, Apr 18 2013
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MATHEMATICA
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CoefficientList[Series[(3*x-1+(1-6*x+7*x^2)/Sqrt[5*x^2-6*x+1])/(2*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 07 2013 *)
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PROG
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(PARI) x='x+O('x^66); Vec((3*x-1+(1-6*x+7*x^2)/sqrt(5*x^2-6*x+1))/(2*x^2)) /* Joerg Arndt, Apr 19 2013 */
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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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A026385
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Sum{T(n-k,k)}, 0<=k<=[ n/2 ], where T is the array in A026374.
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+20
3
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1, 1, 2, 4, 6, 11, 20, 33, 59, 104, 178, 314, 549, 952, 1669, 2913, 5074, 8872, 15482, 27007, 47172, 82325, 143675, 250848, 437822, 764198, 1334041, 2328512, 4064457, 7094833, 12384034, 21616716, 37732990, 65863651
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OFFSET
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0,3
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LINKS
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FORMULA
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G.f.: (1+x+x^2)/(1-x^2-3x^3-x^4). - Ralf Stephan, Apr 30 2004
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MATHEMATICA
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LinearRecurrence[{0, 1, 3, 1}, {1, 1, 2, 4}, 40] (* Harvey P. Dale, Jun 18 2014 *)
CoefficientList[Series[(1 + x + x^2)/(1 - x^2 - 3 x^3 - x^4), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 19 2014 *)
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PROG
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(Magma) I:=[1, 1, 2, 4]; [n le 4 select I[n] else Self(n-2)+3*Self(n-3)+Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jun 19 2014
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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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A026379
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a(n) = number of integer strings s(0),...,s(n) counted by array T in A026374 that have s(n)=3; also a(n) = T(2n-1,n-2).
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+20
2
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1, 7, 39, 202, 1015, 5028, 24731, 121208, 593019, 2899335, 14173401, 69301422, 338990145, 1659037695, 8124085575, 39806373880, 195160896835, 957396540285, 4699409632805, 23080158080150, 113414575414245, 557601196738190
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OFFSET
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2,2
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LINKS
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FORMULA
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a(n) = [t^(n+1)]{(1+t)(1+3t+t^2)^(n-1)}. - Emeric Deutsch, Jan 30 2004
G.f.: -1/x+2-2*x^2/(10*x^2+sqrt(1-5*x)*sqrt(1-x)*(4*x-1)-7*x+1). - Vladimir Kruchinin, Aug 11 2015
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MAPLE
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sum(binomial(n, k)*binomial(2*k+1, k-1), k=0..n); n=0, 1, ... # N. J. A. Sloane
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MATHEMATICA
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Table[Sum[Binomial[n, k]Binomial[2k+1, k-1], {k, 0, n}], {n, 30}] (* Harvey P. Dale, Apr 28 2013 *)
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CROSSREFS
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KEYWORD
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nonn,nice,easy
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AUTHOR
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STATUS
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approved
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1, 3, 4, 11, 17, 45, 75, 195, 339, 873, 1558, 3989, 7247, 18483, 34016, 86515, 160795, 408105, 764388, 1936881, 3650571, 9238023, 17501619, 44241261, 84179877, 212601015, 406020930, 1024642875, 1963073865, 4950790605
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OFFSET
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0,2
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LINKS
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FORMULA
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Davenport et al. give a g.f.
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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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1, 4, 11, 17, 30, 39, 58, 70, 95, 110, 141, 159, 196, 217, 260, 284, 333, 360, 415, 445, 506, 539, 606, 642, 715, 754, 833, 875, 960, 1005, 1096, 1144, 1241, 1292, 1395, 1449, 1558, 1615, 1730, 1790, 1911, 1974, 2101, 2167, 2300
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OFFSET
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2,2
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LINKS
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FORMULA
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G.f.: z^2*(1+3z+5z^2)/[(1-z)^3*(1+z)^2]. - Emeric Deutsch, Jan 25 2004
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MATHEMATICA
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Drop[CoefficientList[Series[z^2(1+3z+5z^2)/((1-z)^3(1+z)^2), {z, 0, 50}], z], 2] (* or *) LinearRecurrence[{1, 2, -2, -1, 1}, {1, 4, 11, 17, 30}, 50] (* Harvey P. Dale, Dec 31 2021 *)
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PROG
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(Haskell)
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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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1, 6, 17, 45, 75, 144, 202, 330, 425, 630, 771, 1071, 1267, 1680, 1940, 2484, 2817, 3510, 3925, 4785, 5291, 6336, 6942, 8190, 8905, 10374, 11207, 12915, 13875, 15840, 16936, 19176, 20417, 22950, 24345, 27189, 28747, 31920
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OFFSET
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3,2
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LINKS
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FORMULA
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G.f.: z^3*(1+5z+8z^2+13z^3)/[(1-z)^4*(1+z)^3]. - Emeric Deutsch, Jan 25 2004
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PROG
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(Haskell)
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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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A026384
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a(n) = Sum_{j=0..i, i=0..n} T(i,j), where T is the array in A026374.
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+20
1
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1, 3, 8, 18, 43, 93, 218, 468, 1093, 2343, 5468, 11718, 27343, 58593, 136718, 292968, 683593, 1464843, 3417968, 7324218, 17089843, 36621093, 85449218, 183105468, 427246093, 915527343, 2136230468, 4577636718, 10681152343, 22888183593, 53405761718
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OFFSET
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0,2
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COMMENTS
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Partial sums of A026383. Number of lattice paths from (0,0) that do not go to right of the line x=n, using the steps U=(1,1), D=(1,-1) and, at levels ...,-4,-2,0,2,4,..., also H=(2,0). Example: a(2)=8 because we have the empty path, U, D, UU, UD, DD, DU and H. - Emeric Deutsch, Feb 18 2004
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LINKS
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FORMULA
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G.f.: (1+2*x) / ((1-x)*(1-5*x^2)). - Ralf Stephan, Apr 30 2004
a(n) = (7*5^(n/2) - 3)/4 for n even.
a(n) = 3*(5^((n+1)/2) - 1)/4 for n odd.
a(n) = a(n-1) + 5*a(n-2) - 5*a(n-3) for n>2.
(End)
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MAPLE
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a[0]:=0:a[1]:=1:for n from 2 to 100 do a[n]:=5*a[n-2]+3 od: seq(a[n], n=1..29); # Zerinvary Lajos, Mar 17 2008
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MATHEMATICA
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CoefficientList[Series[(1 + 2 x) / ((1 - x) (1 - 5 x^2)), {x, 0, 33}], x] (* Vincenzo Librandi, Aug 09 2017 *)
LinearRecurrence[{1, 5, -5}, {1, 3, 8}, 40] (* Harvey P. Dale, May 31 2023 *)
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PROG
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(PARI) Vec((2*x + 1)/(5*x^3 - 5*x^2 - x + 1) + O(x^40)) \\ Colin Barker, Nov 25 2016
(Magma) I:=[1, 3, 8]; [n le 3 select I[n] else Self(n-1)+5*Self(n-2)-5*Self(n-3): n in [1..35]]; // Vincenzo Librandi, Aug 09 2017
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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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STATUS
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approved
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1, 2, 11, 34, 195, 678, 3989, 14494, 86515, 321590, 1936881, 7301142, 44241261, 168359754, 1024642875, 3926147730, 23973456915, 92338836390, 565280386625, 2186194166950, 13411044301945, 52037098259090, 319756851757695, 1244063987615130, 7655279183309725, 29851422385561898
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OFFSET
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0,2
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LINKS
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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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