A120730 -id:A120730 - cp's OEIS frontend

A361887 a(n) = S(5,n), where S(r,n) = Sum_{k = 0..floor(n/2)} ( binomial(n,k) - binomial(n,k-1) )^r.

1, 1, 2, 33, 276, 4150, 65300, 1083425, 20965000, 399876876, 8461219032, 178642861782, 4010820554664, 90684123972156, 2130950905378152, 50560833176021025, 1231721051614138800, 30294218438009039800, 759645100717216142000, 19213764100954274616908, 493269287121905287769776

Offset: 0

Peter Bala, Mar 28 2023

For r a positive integer define S(r,n) = Sum_{k = 0..floor(n/2)} ( binomial(n,k) - binomial(n,k-1) )^r. The present sequence is {S(5,n)}. Gould (1974) conjectured that S(3,n) was always divisible by S(1,n). See A183069 for {S(3,n)/S(1,n)}. In fact, calculation suggests that if r is odd then S(r,n) is always divisible by S(1,n).

a(n) is the total number of 5-tuples of semi-Dyck paths from (0,0) to (n,n-2*j) for j=0..floor(n/2). - Alois P. Heinz, Apr 02 2023

Seiichi Manyama, Table of n, a(n) for n = 0..673
H. W. Gould, Problem E2384, Amer. Math. Monthly, 81 (1974), 170-171.

Cf. A003161 ( S(3,n) ), A003162 ( S(3,n)/S(1,n) ), A183069 ( S(3,2*n-1)/ S(1,2*n-1) ), A361888 ( S(5,n)/S(1,n) ), A361889 ( S(5,2*n-1)/S(1,2*n-1) ), A361890 ( S(7,n) ), A361891 ( S(7,n)/S(1,n) ), A361892 ( S(7,2*n-1)/S(1,2*n-1) ).
Column k=5 of A357824.
Cf. A008315, A120730.

seq(add( ( binomial(n,k) - binomial(n,k-1) )^5, k = 0..floor(n/2)), n = 0..20);

Table[Sum[(Binomial[n, k] - Binomial[n, k-1])^5, {k,0,Floor[n/2]}], {n,0,20}] (* Vaclav Kotesovec, Aug 27 2023 *)

from math import comb
def A361887(n): return sum((comb(n,j)*(m:=n-(j<<1)+1)//(m+j))**5 for j in range((n>>1)+1)) # Chai Wah Wu, Mar 25 2025

a(n) = Sum_{k = 0..floor(n/2)} ( (n - 2*k + 1)/(n - k + 1) * binomial(n,k) )^5.
From Alois P. Heinz, Apr 02 2023: (Start)
a(n) = Sum_{j=0..floor(n/2)} A008315(n,j)^5.
a(n) = Sum_{j=0..n} A120730(n,j)^5.
a(n) = A357824(n,5). (End)
a(n) ~ 2^(5*n + 19/2) / (125 * Pi^(5/2) * n^(9/2)). - Vaclav Kotesovec, Aug 27 2023

A361890 a(n) = S(7,n), where S(r,n) = Sum_{k = 0..floor(n/2)} ( binomial(n,k) - binomial(n,k-1) )^r.

Original entry on oeis.org

1, 1, 2, 129, 2316, 94510, 4939220, 211106945, 14879165560, 828070125876, 61472962084968, 4223017425122958, 325536754765395096, 25399546083773839692, 2059386837863675003112, 173281152533121109073025, 14789443838781868027714800, 1307994690673355979749969800

Offset: 0

Peter Bala, Mar 30 2023

For r a positive integer define S(r,n) = Sum_{k = 0..floor(n/2)} ( binomial(n,k) - binomial(n,k-1) )^r. The present sequence is {S(7,n)}. Gould (1974) conjectured that S(3,n) was always divisible by S(1,n). See A183069 for {S(3,n)/S(1,n)}. In fact, calculation suggests that if r is odd then S(r,n) is always divisible by S(1,n).

a(n) is the total number of 7-tuples of semi-Dyck paths from (0,0) to (n,n-2*j) for j=0..floor(n/2). - Alois P. Heinz, Apr 02 2023

Seiichi Manyama, Table of n, a(n) for n = 0..483
H. W. Gould, Problem E2384, Amer. Math. Monthly, 81 (1974), 170-171.

Cf. A003161 ( S(3,n) ), A003162 ( S(3,n)/S(1,n) ), A382394 ( S(3,2*n-1) ), A183069 ( S(3,2*n-1)/ S(1,2*n-1) ), A361887 ( S(5,n) ), A361888 ( S(5,n)/S(1,n) ), A361889 ( S(5,2*n-1)/S(1,2*n-1) ), A361891 ( S(7,n)/S(1,n) ), A361892 ( S(7,2*n-1)/ S(1,2*n-1) ).
Column k=7 of A357824.
Cf. A008315, A120730.

seq(add( ( binomial(n,k) - binomial(n,k-1) )^7, k = 0..floor(n/2)), n = 0..20);

Table[Sum[(Binomial[n, k] - Binomial[n, k-1])^7, {k,0,Floor[n/2]}], {n,0,20}] (* Vaclav Kotesovec, Aug 27 2023 *)

from math import comb
def A361890(n): return sum((comb(n,j)*(m:=n-(j<<1)+1)//(m+j))**7 for j in range((n>>1)+1)) # Chai Wah Wu, Mar 25 2025

a(n) = Sum_{k = 0..floor(n/2)} ( (n - 2*k + 1)/(n - k + 1) * binomial(n,k) )^7.
From Alois P. Heinz, Apr 02 2023: (Start)
a(n) = Sum_{j=0..floor(n/2)} A008315(n,j)^7.
a(n) = Sum_{j=0..n} A120730(n,j)^7.
a(n) = A357824(n,7). (End)
a(n) ~ 3 * 2^(7*n + 27/2) / (2401 * Pi^(7/2) * n^(13/2)). - Vaclav Kotesovec, Aug 27 2023

A126087 Expansion of c(2x^2)/(1-xc(2x^2)), where c(x) = (1-sqrt(1-4x))/(2*x) is the g.f. of the Catalan numbers (A000108).

Original entry on oeis.org

1, 1, 3, 5, 15, 29, 87, 181, 543, 1181, 3543, 7941, 23823, 54573, 163719, 381333, 1143999, 2699837, 8099511, 19319845, 57959535, 139480397, 418441191, 1014536117, 3043608351, 7426790749, 22280372247, 54669443141, 164008329423

Offset: 0

Philippe Deléham, Mar 03 2007

nonn

Series reversion of x*(1+x)/(1+2*x+3*x^2) [offset 0]. - Paul Barry, Mar 13 2007

Hankel transform is 2^C(n+1,2). - Philippe Deléham, Mar 16 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Alin Bostan, Computer Algebra for Lattice Path Combinatorics, Séminaire de Combinatoire Ph. Flajolet, March 28 2013.
Alin Bostan, Andrew Elvey Price, Anthony John Guttmann, and Jean-Marie Maillard, Stieltjes moment sequences for pattern-avoiding permutations, arXiv:2001.00393 [math.CO], 2020.

Cf. A000108, A089022, A120730.

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (1-Sqrt(1-8*x^2))/(x*(4*x-1+Sqrt(1-8*x^2))) )); // G. C. Greubel, Nov 07 2022

c:=x->(1-sqrt(1-4*x))/2/x: G:=c(2*x^2)/(1-x*c(2*x^2)): Gser:=series(G,x=0,35): seq(coeff(Gser,x,n),n=0..32); # Emeric Deutsch, Mar 04 2007

CoefficientList[Series[(1-Sqrt[1-8*x^2])/(x*(4*x-1+Sqrt[1-8*x^2])), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 13 2014 *)

def A120730(n, k): return 0 if (n>2*k) else binomial(n, k)*(2*k-n+1)/(k+1)
def A126087(n): return sum(2^(n-k)*A120730(n,k) for k in range(n+1))
[A126087(n) for n in range(51)] # G. C. Greubel, Nov 07 2022

G.f.: (1-sqrt(1-8*x^2))/(x*(4*x-1+sqrt(1-8*x^2))). - Emeric Deutsch, Mar 04 2007
a(n) = Sum_{k=0..n} 2^(n-k)*A120730(n,k). - Philippe Deléham, Oct 16 2008
a(n-1) = Sum_{k=1..n} (1+(-1)^(n-k))*k*2^((n-k)/2-1)*C(n,floor((n+k)/2))/n. - Vladimir Kruchinin, Feb 18 2011
a(2*n) = A089022(n). - Philippe Deléham, Nov 02 2011
D-finite with recurrence: (n+1)*a(n) = 3*(n+1)*a(n-1) - 8*(2-n)*a(n-2) - 24*(n-2)*a(n-3). - R. J. Mathar, Nov 14 2011
a(n) ~ 2^(3*(n+1)/2) * (3+2*sqrt(2) + (3-2*sqrt(2))*(-1)^n) / (n^(3/2) * sqrt(Pi)). - Vaclav Kotesovec, Feb 13 2014

More terms from Emeric Deutsch, Mar 04 2007

A151254 Number of walks within N^3 (the first octant of Z^3) starting at (0,0,0) and consisting of n steps taken from {(-1, 0, 0), (1, 0, 0), (1, 0, 1), (1, 1, 0), (1, 1, 1)}.

Original entry on oeis.org

1, 4, 20, 96, 480, 2368, 11840, 58880, 294400, 1468416, 7342080, 36667392, 183336960, 916144128, 4580720640, 22896574464, 114482872320, 572320645120, 2861603225600, 14306741583872, 71533707919360, 357650927714304, 1788254638571520, 8941026626502656, 44705133132513280, 223522175800311808

Offset: 0

Manuel Kauers, Nov 18 2008

Hankel transform is 4^binomial(n+1,2). - Philippe Deléham, Feb 01 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Paul Barry, A Note on a One-Parameter Family of Catalan-Like Numbers, JIS 12 (2009) 09.5.4
Alin Bostan and Manuel Kauers, Automatic Classification of Restricted Lattice Walks, arXiv:0811.2899 [math.CO], 2009.

Cf. A000108, A001405, A120730, A151162, A151281, A151403, A156058.

[n le 3 select Factorial(n+2)/6 else (5*n*Self(n-1) + 16*(n-3)*Self(n-2) - 80*(n-3)*Self(n-3))/n: n in [1..30]]; // G. C. Greubel, Nov 09 2022

aux[i_, j_, k_, n_]:= Which[Min[i, j, k, n]<0 || Max[i, j, k]>n, 0, n==0, KroneckerDelta[i, j, k, n], True, aux[i, j, k, n] = aux[-1+i, -1+j, -1+k, -1+n] + aux[-1+i, -1+j, k, -1+n] + aux[-1+i, j, -1+k, -1+n] + aux[-1+i, j, k, -1 + n] + aux[1+i, j, k, -1+n]]; Table[Sum[aux[i,j,k,n], {i,0,n}, {j,0,n}, {k,0,n}], {n, 0, 30}]
a[n_]:= a[n]= If[n<3, (n+3)!/3!, (5*(n+1)*a[n-1] +16*(n-2)*a[n-2] -80*(n-2)*a[n- 3])/(n+1)]; Table[a[n], {n, 0, 30}] (* G. C. Greubel, Nov 09 2022 *)

def a(n): # a = A151254
    if (n==0): return 1
    elif (n%2==1): return 5*a(n-1) - 4^((n-1)/2)*catalan_number((n-1)/2)
    else: return 5*a(n-1)
[a(n) for n in (0..30)] # G. C. Greubel, Nov 09 2022

a(n) = Sum_{k=0..n} A120730(n,k)*4^k. - Philippe Deléham, Feb 01 2009
From Philippe Deléham, Feb 02 2009: (Start)
a(2n+2) = 5*a(2n+1), a(2n+1) = 5*a(2n) - 4^n*A000108(n) = 5*a(2n) - A151403(n).
G.f.: (sqrt(1-16*x^2) + 8*x - 1)/(8*x*(1-5*x)). (End)
a(n) = (5*(n+1)*a(n-1) + 16*(n-2)*a(n-2) - 80*(n-2)*a(n-3))/(n+1). - G. C. Greubel, Nov 09 2022

A357824 Total number A(n,k) of k-tuples of semi-Dyck paths from (0,0) to (n,n-2*j) for j=0..floor(n/2); square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 3, 3, 1, 1, 2, 5, 6, 3, 1, 1, 2, 9, 14, 10, 4, 1, 1, 2, 17, 36, 42, 20, 4, 1, 1, 2, 33, 98, 190, 132, 35, 5, 1, 1, 2, 65, 276, 882, 980, 429, 70, 5, 1, 1, 2, 129, 794, 4150, 7812, 5705, 1430, 126, 6, 1, 1, 2, 257, 2316, 19722, 65300, 78129, 33040, 4862, 252, 6

Offset: 0

Alois P. Heinz, Oct 14 2022

			Square array A(n,k) begins:
  1,  1,   1,    1,     1,       1,        1,         1, ...
  1,  1,   1,    1,     1,       1,        1,         1, ...
  2,  2,   2,    2,     2,       2,        2,         2, ...
  2,  3,   5,    9,    17,      33,       65,       129, ...
  3,  6,  14,   36,    98,     276,      794,      2316, ...
  3, 10,  42,  190,   882,    4150,    19722,     94510, ...
  4, 20, 132,  980,  7812,   65300,   562692,   4939220, ...
  4, 35, 429, 5705, 78129, 1083425, 15105729, 211106945, ...

Alois P. Heinz, Antidiagonals n = 0..121, flattened
Wikipedia, Counting lattice paths

Columns k=0-7 give A008619, A001405, A000108, A003161, A129123, A361887, A382433, A361890.
Rows n=1-5 give: A000012, A007395, A000051, A001550, A074511.
Main diagonal gives A357825.
Cf. A008315, A120730.

b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
     `if`(x=0, 1, add(b(x-1, y+j), j=[-1, 1])))
    end:
A:= (n, k)-> add(b(n, n-2*j)^k, j=0..n/2):
seq(seq(A(n, d-n), n=0..d), d=0..12);

b[x_, y_] := b[x, y] = If[y < 0 || y > x, 0, If[x == 0, 1, Sum[b[x - 1, y + j], {j, {-1, 1}}]]];
A[n_, k_] := Sum[b[n, n - 2*j]^k, { j, 0, n/2}];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Oct 18 2022, after Alois P. Heinz *)

A(n,k) = Sum_{j=0..floor(n/2)} A008315(n,j)^k.

A(n,k) = Sum_{j=0..n} A120730(n,j)^k for k>=1, A(n,0) = A008619(n).

A121724 Generalized central binomial coefficients for k=2.

Original entry on oeis.org

1, 1, 5, 9, 45, 97, 485, 1145, 5725, 14289, 71445, 185193, 925965, 2467137, 12335685, 33563481, 167817405, 464221105, 2321105525, 6507351113, 32536755565, 92236247841, 461181239205, 1319640776249, 6598203881245, 19031570387857, 95157851939285

Offset: 0

Paul Barry, Aug 17 2006, Feb 28 2007

Hankel transform is 4^binomial(n+1,2) = A053763(n+1). Case k=2 of T(n,k) = (1/Pi)*2*k^2*(2*k)^n*Integral_{x=-1..1} x^n*sqrt(1-x^2)/(1+k^2-2*k*x) dx. T(n,k) has Hankel transform (k^2)^binomial(n+1,2). k=1 corresponds to C(n,floor(n/2)).

Series reversion of x*(1+x)/(1+2*x+5*x^2).

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000108, A009766, A053763, A120730.

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (Sqrt(1-16*x^2)+2*x-1)/(2*x*(1-5*x)) )); // G. C. Greubel, Nov 07 2022

CoefficientList[Series[(Sqrt[1-16*x^2]+2*x-1)/(2*x*(1-5*x)), {x,0,40}], x] (* Vaclav Kotesovec, Feb 13 2014 *)

def A120730(n, k): return 0 if (n>2*k) else binomial(n, k)*(2*k-n+1)/(k+1)
def A121724(n): return sum(4^(n-k)*A120730(n,k) for k in range(n+1))
[A121724(n) for n in range(51)] # G. C. Greubel, Nov 07 2022

G.f.: (sqrt(1-16*x^2) + 2*x - 1)/(2*x*(1-5*x)) = c(4*x^2)/(1-x*c(4*x^2)), c(x) the g.f. of A000108.
a(n) = (1/(n+1))*Sum_{k=0..n+1} Sum_{j=0..k} C(n,k)*C(k,j)*C(2*n-2*k+j, n-2*k+j)*(-1)^(n-2*k+j)*2^j*5^(k-j).
a(n) = (1/Pi)*8*4^n*Integral_{x=-1..1} x^n*sqrt(1-x^2)/(5-4*x) dx.
a(n) = Sum_{k=0..floor(n/2)} A009766(n-k,k)*2^2k. - Philippe Deléham, Aug 18 2006
a(n) = Sum_{k=0..n} 4^(n-k)*A120730(n,k). - Philippe Deléham, Oct 16 2008
Conjecture: (n+1)*a(n) = 5*(n+1)*a(n-1) + 16*(n-2)*a(n-2) - 80*(n-2)*a(n-3). - R. J. Mathar, Nov 26 2012
a(n) ~ (9+(-1)^n) * 2^(2*n+5/2) / (9 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 13 2014

More terms from Vincenzo Librandi, Feb 15 2014

A128386 Expansion of c(3x^2)/(1-xc(3*x^2)), c(x) the g.f. of A000108.

Original entry on oeis.org

1, 1, 4, 7, 28, 58, 232, 523, 2092, 4966, 19864, 48838, 195352, 492724, 1970896, 5068915, 20275660, 52955950, 211823800, 560198962, 2240795848, 5987822380, 23951289520, 64563867454, 258255469816, 701383563388, 2805534253552

Offset: 0

Paul Barry, Feb 28 2007

Hankel transform is 3^C(n+1,2) = A047656(n+1).

Series reversion of x*(1+x)/(1+2*x+4*x^2).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Alin Bostan, Computer Algebra for Lattice Path Combinatorics, Slides, Séminaire de Combinatoire Ph. Flajolet, March 28 2013.
Alin Bostan, Andrew Elvey Price, Anthony John Guttmann, and Jean-Marie Maillard, Stieltjes moment sequences for pattern-avoiding permutations, arXiv:2001.00393 [math.CO], 2020.

Cf. A000108, A009766, A047656, A120730.

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (Sqrt(1-12*x^2)+2*x-1)/(2*x*(1-4*x)) )); // G. C. Greubel, Nov 07 2022

A120730[n_, k_]:= If[n>2*k, 0, Binomial[n,k]*(2*k-n+1)/(k+1)];
A126386[n_]:= Sum[3^k*A120730[n, n-k], {k,0,n}];
Table[A126386[n], {n,0,50}] (* G. C. Greubel, Nov 07 2022 *)

def A120730(n, k): return 0 if (n>2*k) else binomial(n, k)*(2*k-n+1)/(k+1)
def A126386(n): return sum(3^k*A120730(n,n-k) for k in range(n+1))
[A126386(n) for n in range(51)] # G. C. Greubel, Nov 07 2022

G.f.: (sqrt(1-12*x^2)+2*x-1)/(2*x*(1-4*x)).
a(n) = (1/(n+1))*Sum_{k=0..n+1} Sum_{j=0..k} C(n,k)*C(k,j)*C(2*n-2*k+j, n-2*k+j)*(-1)^(n+j)*2^(2*k-j).
a(n) = Sum_{k=0..floor(n/2)} C(n,n-k)*(n-2*k+1)*3^k/(n-k+1);
a(n) = Sum_{k=0..floor(n/2)} A009766(n-k,k)*3^k.
a(n) = Sum_{k=0..n} 3^k*A120730(n,n-k). - Philippe Deléham, Mar 03 2007
D-finite with recurrence (n+1)*a(n) - 4*(n+1)*a(n-1) + 12*(2-n)*a(n-2) + 48*(n-2)*a(n-3) = 0. - R. J. Mathar, Nov 14 2011

A128387 Expansion of c(5x^2)/(1-x*c(5x^2)), where c(x) is the g.f. of A000108.

Original entry on oeis.org

1, 1, 6, 11, 66, 146, 876, 2131, 12786, 32966, 197796, 530526, 3183156, 8786436, 52718616, 148733571, 892401426, 2561439806, 15368638836, 44731364266, 268388185596, 790211926076, 4741271556456, 14095578557486

Offset: 0

Paul Barry, Feb 28 2007

Hankel transform is 5^C(n+1,2).

Reversion of x*(1+x)/(1+2*x+6*x^2).

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000108, A009766, A120730, A126386.

R:=PowerSeriesRing(Rationals(), 50); Coefficients(R!( (Sqrt(1-20*x^2)+2*x-1)/(2*x*(1-6*x)) )); // G. C. Greubel, Nov 07 2022

A120730[n_, k_]:= If[n>2*k, 0, Binomial[n, k]*(2*k-n+1)/(k+1)];
A126387[n_]:= Sum[5^k*A120730[n, n-k], {k,0,n}];
Table[A126387[n], {n, 0, 50}] (* G. C. Greubel, Nov 07 2022 *)

def A120730(n, k): return 0 if (n>2*k) else binomial(n, k)*(2*k-n+1)/(k+1)
def A126387(n): return sum(5^k*A120730(n,n-k) for k in range(n+1))
[A126387(n) for n in range(51)] # G. C. Greubel, Nov 07 2022

G.f.: (sqrt(1-20*x^2) + 2*x - 1)/(2*x*(1-6*x)).
a(n) = (1/(n+1))*Sum_{k=0..n+1} Sum_{j=0..k} C(n,k)*C(k,j)*C(2*n-2*k+j, n-2*k+j)*(-1)^(n+j)*2^j*6^(k-j).
a(n) = Sum_{k=0..floor(n/2)} C(n,n-k)*(n-2*k+1)*5^k/(n-k+1).
a(n) = Sum_{k=0..floor(n/2)} A009766(n-k,k)*5^k.
a(n) = Sum_{k=0..n} 5^k*A120730(n,n-k). - Philippe Deléham, Mar 03 2007
(n+1)*a(n) = 6*(n+1)*a(n-1) + 20*(n-2)*a(n-2) - 120*(n-2)*a(n-3). - R. J. Mathar, Nov 14 2011

A156195 a(2n+2) = 6a(2n+1), a(2n+1) = 6a(2n) - 5^n*A000108(n), a(0)=1.

Original entry on oeis.org

1, 5, 30, 175, 1050, 6250, 37500, 224375, 1346250, 8068750, 48412500, 290343750, 1742062500, 10450312500, 62701875000, 376177734375, 2257066406250, 13541839843750, 81251039062500, 487496738281250, 2924980429687500, 17549718554687500, 105298311328125000

Offset: 0

Philippe Deléham, Feb 05 2009

nonn

Hankel transform is 5^C(n+1,2). - Philippe Deléham, Feb 05 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Paul Barry, A Note on a One-Parameter Family of Catalan-Like Numbers, JIS 12 (2009) 09.5.4.

Cf. A000108, A001405, A120730, A151162, A151254, A151281, A156058.

[n le 3 select Factorial(n+3)/24 else (6*n*Self(n-1) + 20*(n-3)*Self(n-2) - 120*(n-3)*Self(n-3))/n: n in [1..30]]; // G. C. Greubel, Nov 09 2022

A156195 := proc(n)
    option remember;
    local nh;
    if n= 0 then
        1;
    elif  type(n,'even') then
        6*procname(n-1);
    else
        nh := floor(n/2) ;
        6*procname(n-1)-5^nh*A000108(nh) ;
    end if;
end proc: # R. J. Mathar, Jul 21 2016

CoefficientList[Series[(Sqrt[1-20x^2]+10x-1)/(10x(1-6x)),{x,0,30}],x] (* Harvey P. Dale, Oct 21 2016 *)

def a(n): # a = A156195
    if (n==0): return 1
    elif (n%2==1): return 6*a(n-1) - 5^((n-1)/2)*catalan_number((n-1)/2)
    else: return 6*a(n-1)
[a(n) for n in (0..30)] # G. C. Greubel, Nov 09 2022

a(n) = Sum_{k=0..n} A120730(n,k)*5^k.
G.f.: (sqrt(1-20*x^2) +10*x -1)/(10*x*(1-6*x)). - Philippe Deléham, Feb 05 2009
(n+1)*a(n) = 6*(n+1)*a(n-1) + 20*(n-2)*a(n-2) - 120*(n-2)*a(n-3). - R. J. Mathar, Jul 21 2016

Corrected and extended by Harvey P. Dale, Oct 21 2016

A382433 a(n) = S(6,n), where S(r,n) = Sum_{k=0..floor(n/2)} ( binomial(n,k) - binomial(n,k-1) )^r.

Original entry on oeis.org

1, 1, 2, 65, 794, 19722, 562692, 15105729, 553537490, 18107304842, 716747344436, 27247858130506, 1137502720488532, 47573235297987700, 2085487143991309320, 92820152112054862785, 4246321874111740074210, 197525644801830489637170, 9363425291004877645851300

Offset: 0

Seiichi Manyama, Mar 25 2025

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..562

Column k=6 of A357824.
Cf. A000108, A129123.
Cf. A008315, A120730, A382435.

b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
     `if`(x=0, 1, add(b(x-1, y+j), j=[-1, 1])))
    end:
a:= n-> add(b(n, n-2*j)^6, j=0..n/2):
seq(a(n), n=0..18);  # Alois P. Heinz, Mar 25 2025

Table[Sum[Binomial[n,k] * (Binomial[n,k] - Binomial[n,k-1])^5, {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Mar 25 2025 *)

a(n) = sum(k=0, n, binomial(n, k)*(binomial(n, k)-binomial(n, k-1))^5);

from math import comb
def A382433(n): return sum((comb(n,j)*(m:=n-(j<<1)+1)//(m+j))**6 for j in range((n>>1)+1)) # Chai Wah Wu, Mar 25 2025

a(n) = Sum_{k=0..floor(n/2)} A008315(n,k)^6.
a(n) = Sum_{k=0..n} A120730(n,k)^6.
a(n) = A357824(n,6).
a(n) = Sum_{k=0..n} binomial(n,k) * ( binomial(n,k) - binomial(n,k-1) )^5.
a(n) ~ 5 * 2^(6*n+4) / (3^(5/2) * Pi^(5/2) * n^(11/2)). - Vaclav Kotesovec, Mar 25 2025

cp's OEIS Frontend

A361887 a(n) = S(5,n), where S(r,n) = Sum_{k = 0..floor(n/2)} ( binomial(n,k) - binomial(n,k-1) )^r.

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A361890 a(n) = S(7,n), where S(r,n) = Sum_{k = 0..floor(n/2)} ( binomial(n,k) - binomial(n,k-1) )^r.

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A126087 Expansion of c(2*x^2)/(1-x*c(2*x^2)), where c(x) = (1-sqrt(1-4*x))/(2*x) is the g.f. of the Catalan numbers (A000108).

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A151254 Number of walks within N^3 (the first octant of Z^3) starting at (0,0,0) and consisting of n steps taken from {(-1, 0, 0), (1, 0, 0), (1, 0, 1), (1, 1, 0), (1, 1, 1)}.

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A357824 Total number A(n,k) of k-tuples of semi-Dyck paths from (0,0) to (n,n-2*j) for j=0..floor(n/2); square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A121724 Generalized central binomial coefficients for k=2.

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A128386 Expansion of c(3*x^2)/(1-x*c(3*x^2)), c(x) the g.f. of A000108.

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A126087 Expansion of c(2x^2)/(1-xc(2x^2)), where c(x) = (1-sqrt(1-4x))/(2*x) is the g.f. of the Catalan numbers (A000108).

A128386 Expansion of c(3x^2)/(1-xc(3*x^2)), c(x) the g.f. of A000108.

A156195 a(2n+2) = 6a(2n+1), a(2n+1) = 6a(2n) - 5^n*A000108(n), a(0)=1.