A002893 -id:A002893 - cp's OEIS frontend

A169711 The function W_n(6) (see Borwein et al. reference for definition).

1, 20, 93, 256, 545, 996, 1645, 2528, 3681, 5140, 6941, 9120, 11713, 14756, 18285, 22336, 26945, 32148, 37981, 44480, 51681, 59620, 68333, 77856, 88225, 99476, 111645, 124768, 138881, 154020, 170221, 187520, 205953, 225556, 246365, 268416, 291745, 316388

Offset: 1

N. J. A. Sloane, Apr 17 2010

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Jonathan M. Borwein, Dirk Nuyens, Armin Straub, and James Wan, Some Arithmetic Properties of Short Random Walk Integrals, May 2011.
Pakawut Jiradilok and Elchanan Mossel, Gaussian Broadcast on Grids, arXiv:2402.11990 [cs.IT], 2024. See p. 27.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

The sequence in Table 1 of the Borwein et al. reference are A000384, A109711-A109713; A000984, A002893, A002895, A169714, A169715.
Column 3 of A287316.
Cf. A287314.

[6*n^3-9*n^2+4*n: n in [1..40]]; // Vincenzo Librandi, May 28 2017

A169711 := proc(n)
        W(n,6) ;
end proc:
seq(A169711(n),n=1..20) ; # uses W from A169715; R. J. Mathar, Mar 28 2012
a := n -> 6*n^3 - 9*n^2 + 4*n: seq(a(n), n=1..33); # Peter Luschny, May 27 2017

CoefficientList[Series[(1 + 16 x + 19 x^2) / (1 - x)^4, {x, 0, 50}], x] (* or *) Table[6 n^3 - 9 n^2 + 4 n, {n, 1, 40}] (* Vincenzo Librandi, May 28 2017 *)
LinearRecurrence[{4,-6,4,-1},{1,20,93,256},40] (* Harvey P. Dale, Feb 27 2023 *)

a(n)=6*n^3-9*n^2+4*n \\ Charles R Greathouse IV, Oct 18 2022

a(n) = 6*n^3 - 9*n^2 + 4*n. - Peter Luschny, May 27 2017
G.f.: x*(1+16*x+19*x^2)/(1-x)^4. - Vincenzo Librandi, May 28 2017
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, May 28 2017

A248586 a(n) = Sum_{i=0..n} C(n,i)*C(2i,i)^2.

Original entry on oeis.org

1, 5, 45, 521, 6733, 92385, 1316865, 19274925, 287694285, 4359037985, 66837293545, 1034774126325, 16149186405025, 253737607849445, 4009771017244485, 63681603585696321, 1015763347140335565, 16264070907887454465

Offset: 0

R. J. Mathar, Oct 09 2014

Seiichi Manyama, Table of n, a(n) for n = 0..815

Cf. A002894 (inverse binomial transform), A002893.

Table[Sum[Binomial[n, k] Binomial[2k, k]^2, {k, 0, n}],{n,0,100}] (* Emanuele Munarini, Oct 28 2016 *)

makelist(sum(binomial(n,k)*binomial(2*k,k)^2,k,0,n),n,0,12); /* Emanuele Munarini, Oct 28 2016 */

a(n) = sum(i=0, n, binomial(n,i)*binomial(2*i,i)^2); \\ Michel Marcus, Oct 09 2014

a(n) = Sum_{i=0..n} A007318(n,i)*A002894(i).
Conjecture: n^2*a(n) +(-19*n^2+19*n-5)*a(n-1) +35*(n-1)^2*a(n-2) -17*(n-1)*(n-2)*a(n-3)=0.
G.f.: LegendreP(-1/2, (1+15x)/(1-17x)) /[sqrt(1-17x)*sqrt(1-x)]. [Corrected by Robert Israel, Oct 28 2016]
From Emanuele Munarini, Oct 28 2016: (Start)
a(n) = hypergeometric(1/2,1/2,-n;1,1;-16).
G.f.: A(t) = (2/Pi)*(ellipticK(16*t/(1-t))/(1-t)).
Diff. eq. satisfied by the g.f.: t*(1-t)*(1-18*t+17*t^2)*A''(t)+(1-t)*(1-37*t+68*t^2)*A'(t)-(34*t^2-35*t+5)*A(t)=0.
Remark: the conjectured recurrence for the coefficients a(n) comes from this diff. eq. for A(t).
(End)
a(n) ~ 17^(n+1)/(16*Pi*n). - Vaclav Kotesovec, Oct 30 2016

A336270 a(n) = Sum_{k=0..n} Sum_{j=0..k} (binomial(n,k) * binomial(k,j))^n.

Original entry on oeis.org

1, 3, 15, 381, 67635, 83118753, 813824623689, 58040410068847251, 32150480245981639533315, 154935057570894645075940703673, 5474671509704049919709361235659936825, 1600436120524545216094358662984789029130593831

Offset: 0

Ilya Gutkovskiy, Jul 15 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..47

Cf. A000244, A002893, A141057, A167010, A172434, A180350.

Table[Sum[Sum[(Binomial[n, k] Binomial[k, j])^n, {j, 0, k}], {k, 0, n}], {n, 0, 11}]
Table[(n!)^n SeriesCoefficient[Sum[x^k/(k!)^n, {k, 0, n}]^3, {x, 0, n}], {n, 0, 11}]

a(n) = sum(k=0, n, sum(j=0, k, (binomial(n,k) * binomial(k,j))^n)); \\ Michel Marcus, Jul 16 2020

a(n) = (n!)^n * [x^n] (Sum_{k>=0} x^k / (k!)^n)^3.

A336271 a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k+1) * binomial(n,k)^2 * binomial(2k,k) a(n-k).

Original entry on oeis.org

1, 2, 10, 92, 1354, 29252, 873964, 34555880, 1748176714, 110183215988, 8467704986260, 779536758060920, 84699429189141100, 10725613123706081720, 1565870044943751242440, 261092436660169105108592, 49312362996510562406915914, 10473104312824253527997052500

Offset: 0

Ilya Gutkovskiy, Jul 15 2020

nonn

Cf. A000275, A000984, A002893.

Column k=2 of A340986.

a[0] = 1; a[n_] := a[n] = Sum[(-1)^(k + 1) Binomial[n, k]^2 Binomial[2 k, k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 17}]
nmax = 17; CoefficientList[Series[1/BesselJ[0, 2 Sqrt[x]]^2, {x, 0, nmax}], x] Range[0, nmax]!^2

Sum_{n>=0} a(n) * x^n / (n!)^2 = 1 / BesselJ(0,2*sqrt(x))^2.

a(n) ~ (n!)^2 * n / (BesselJ(1, 2*sqrt(r))^2 * r^(n+1)), where r = BesselJZero(0,1)^2 / 4 = A115368^2/4 = 1.4457964907366961302939989396139517587... - Vaclav Kotesovec, Jul 15 2020

A172434 G.f.: Sum_{n>=0} a(n)*x^n/n!^4 = [ Sum_{n>=0} x^n/n!^4 ]^3.

Original entry on oeis.org

1, 3, 51, 1785, 67635, 2973753, 146591529, 7735733883, 430208938035, 24954576411225, 1496639801457801, 92241539987122683, 5816057121183700521, 373854785336483200155, 24431647104881328618315, 1619654401178752389082785

Offset: 0

Paul D. Hanna, Jan 20 2011

nonn

			G.f.: A(x) = 1 + 3*x + 51*x^2/2!^4 + 1785*x^3/3!^4 + 67635*x^4/4!^4 +...
A(x)^(1/3) = 1 + x + x^2/2!^4 + x^3/3!^4 + x^4/4!^4 +...

Seiichi Manyama, Table of n, a(n) for n = 0..500

Cf. A002893, A005260, A141057, A180350, A336270.

{a(n)=if(n<0, 0, n!^4*polcoeff(sum(m=0, n, x^m/m!^4+x*O(x^n))^3, n))}

{a(n)=sum(k=0, n, binomial(n, k)^4*sum(j=0, k, binomial(k, j)^4))}

a(n) = Sum_{k=0..n} C(n,k)^4 * Sum_{j=0..k} C(k,j)^4 = Sum_{k=0..n} C(n,k)^4 * A005260(k).

A180350 G.f.: Sum_{n>=0} a(n)*x^n/n!^5 = [ Sum_{n>=0} x^n/n!^5 ]^3.

Original entry on oeis.org

1, 3, 99, 9237, 775971, 83118753, 10657602909, 1463886204147, 215566192274211, 33677584957306713, 5492032622227428849, 928229455634614797447, 161727023896151286167901, 28905146810167510775300463

Offset: 0

Paul D. Hanna, Jan 20 2011

nonn

			G.f.: A(x) = 1 + 3*x + 99*x^2/2!^5 + 9237*x^3/3!^5 + 775971*x^4/4!^5 +...
A(x)^(1/3) = 1 + x + x^2/2!^5 + x^3/3!^5 + x^4/4!^5 +...

Seiichi Manyama, Table of n, a(n) for n = 0..424

Cf. A002893, A005261, A141057, A172434, A336270.

{a(n)=if(n<0, 0, n!^5*polcoeff(sum(m=0, n, x^m/m!^5+x*O(x^n))^3, n))}

{a(n)=sum(k=0, n, binomial(n, k)^5*sum(j=0, k, binomial(k, j)^5))}

a(n) = Sum_{k=0..n} C(n,k)^5 * Sum_{j=0..k} C(k,j)^5 = Sum_{k=0..n} C(n,k)^5 * A005261(k).

A186377 a(n) equals the least sum of the squares of the coefficients in (1 + 2*x^k + x^p + x^q)^n found at sufficiently large p and q>(n+1)p for some fixed k>0.

Original entry on oeis.org

1, 7, 79, 1129, 18559, 333577, 6365089, 126652183, 2598628543, 54577439833, 1167481074529, 25346459683783, 557042221952881, 12368307313680871, 277027947337574911, 6251808554314780009, 142015508983550880703

Offset: 0

Paul D. Hanna, Feb 19 2011

nonn

Equivalently, a(n) equals the sum of the squares of the coefficients in any one of the following polynomials:
. (2 + x^k + x^p + x^q)^n, or
. (1 + x^k + 2*x^p + x^q)^n, or
. (1 + x^k + x^p + 2*x^q)^n,
for all p>(n+1)k and q>(n+1)p and fixed k>0.

			G.f.: A(x) = 1 + 7*x + 79*x^2/2!^2 + 1129*x^3/3!^2 + 18559*x^4/4!^2 +...
The g.f. may be expressed as:
A(x) = [Sum_{n>=0} x^n/n!^2]^3 *[Sum_{n>=0} (4x)^n/n!^2] where
[Sum_{n>=0} x^n/n!^2]^3 = 1 + 3*x + 15*x^2/2!^2 + 93*x^3/3!^2 + 639*x^4/4!^2 + 4653*x^5/5!^2 +...+ A002893(n)*x^n/n!^2 +...

Cf. A186375, A186376, A186378.

Table[Sum[Binomial[n,k]^2 * 4^(n-k) *Sum[Binomial[k,j]^2 * Binomial[2j,j], {j,0,k}], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Feb 11 2015 *)

{a(n)=local(V=Vec((1+2*x+x^(n+2)+x^(n^2+2*n+3))^n));V*V~}

{a(n)=sum(k=0,n,binomial(n,k)^2*4^(n-k)*sum(j=0,k,binomial(k,j)^2*binomial(2*j,j)))}

{a(n)=n!^2*polcoeff(sum(m=0,n,x^m/m!^2)^3*sum(m=0,n,(2^2*x)^m/m!^2),n)}

(1) a(n) = Sum_{k=0..n} C(n,k)^2 *4^(n-k) *Sum_{j=0..k} C(k,j)^2*C(2j,j).
Let g.f. A(x) = Sum_{n>=0} a(n)*x^n/n!^2, then
(2) A(x) = B(x)^3 * B(2^2*x)
where B(x) = Sum_{n>=0} x^n/n!^2 = BesselI(0, 2*sqrt(x)).
Recurrence: (n-1)*n^3*(3*n - 5)*a(n) = 2*(n-1)*(54*n^4 - 174*n^3 + 192*n^2 - 99*n + 20)*a(n-1) - 2*(441*n^5 - 2604*n^4 + 6102*n^3 - 7107*n^2 + 4111*n - 940)*a(n-2) + 2*(n-2)^2*(726*n^3 - 3076*n^2 + 4188*n - 1655)*a(n-3) - 225*(n-3)^2*(n-2)^2*(3*n - 2)*a(n-4). - Vaclav Kotesovec, Feb 12 2015
a(n) ~ 5^(2*n+2) / (2^(7/2) * Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Feb 12 2015

A336638 Sum_{n>=0} a(n) * x^n / (n!)^2 = 1 / BesselJ(0,2*sqrt(x))^3.

Original entry on oeis.org

1, 3, 21, 255, 4725, 123903, 4368729, 199467243, 11455187445, 808475761695, 68805857523321, 6950458374996843, 822292004658568761, 112639503374757412875, 17688916392275574761805, 3157133540377493872350855, 635546443798928578953138165

Offset: 0

Ilya Gutkovskiy, Jul 28 2020

nonn

Cf. A000275, A002893, A336271, A336639.

Column k=3 of A340986.

nmax = 16; CoefficientList[Series[1/BesselJ[0, 2 Sqrt[x]]^3, {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = 1; a[n_] := a[n] = Sum[(-1)^(k + 1) Binomial[n, k]^2 HypergeometricPFQ[{1/2, -k, -k}, {1, 1}, 4] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 16}]

a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k+1) * binomial(n,k)^2 * A002893(k) * a(n-k).

a(n) ~ n!^2 * n^2 / (2 * r^(n + 3/2) * BesselJ(1, 2*sqrt(r))^3), where r = BesselJZero(0,1)^2 / 4 = A115368^2/4 = 1.4457964907366961302939989396139517587... - Vaclav Kotesovec, Jul 11 2025

A208426 Expansion of Sum_{n>=0} (3n)!/n!^3 x^(2n)/(1-3x)^(3*n+1).

Original entry on oeis.org

1, 3, 15, 99, 711, 5373, 42099, 338355, 2771127, 23028813, 193610385, 1643215005, 14056350075, 121040308665, 1048212778635, 9122168556819, 79727173530327, 699443806767525, 6156776010386481, 54356715121718349, 481194980656865721, 4270165015550478003

Offset: 0

Paul D. Hanna, Feb 26 2012

nonn

Compare g.f. to: Sum_{n>=0} (3*n)!/n!^3 * x^(2*n)/(1-2*x)^(3*n+1), which is a g.f. of the Franel numbers (A000172).
Diagonal of rational functions 1/(1 - x*y - y*z - x*z - 3*x*y*z), 1/(1 - x*y + y*z + x*z - 3*x*y*z). - Gheorghe Coserea, Jul 04 2018
Diagonal of the rational function 1/(1 - (x^2 + y^2 + z^2 + 3*x*y*z)). - Seiichi Manyama, Jul 05 2025

			G.f.: A(x) = 1 + 3*x + 15*x^2 + 99*x^3 + 711*x^4 + 5373*x^5 + 42099*x^6 + ...
where
A(x) = 1/(1-3*x) + 6*x^2/(1-3*x)^4 + 90*x^4/(1-3*x)^7 + 1680*x^6/(1-3*x)^10 + 34650*x^8/(1-3*x)^13 + 756756*x^10/(1-3*x)^16 + ...

Gheorghe Coserea, Table of n, a(n) for n = 0..200

Cf. A000172, A002893, A069835, A208425, A294035.

Table[3^n * HypergeometricPFQ[{1/2 - n/2, -n/2, 1 + n}, {1, 1}, 4/9], {n, 0, 25}] (* Vaclav Kotesovec, Oct 07 2020 *)

{a(n)=polcoeff(sum(m=0,n, (3*m)!/m!^3*x^(2*m)/(1-3*x+x*O(x^n))^(3*m+1)),n)}
for(n=0,31,print1(a(n),", "))

a(n) = sum(k=0, n\2, (n+k)!/(k!^3*(n-2*k)!) * 3^(n-2*k)); \\ Gheorghe Coserea, Jul 04 2018

From Gheorghe Coserea, Jul 04 2018: (Start)
a(n) = Sum_{k=0..floor(n/2)} (n+k)!/(k!^3*(n-2*k)!) * 3^(n-2*k).
G.f. y=A(x) satisfies: 0 = x*(3*x + 2)*(27*x^3 + 9*x - 1)*y'' + (243*x^4 + 216*x^3 + 27*x^2 + 36*x - 2)*y' + 3*(27*x^3 + 33*x^2 - 2*x + 2)*y.
(End)
From Vaclav Kotesovec, Oct 07 2020: (Start)
Recurrence: n^2*(3*n - 5)*a(n) = 3*(9*n^3 - 24*n^2 + 17*n - 4)*a(n-1) + 3*(3*n - 4)*a(n-2) + 27*(n-2)^2*(3*n - 2)*a(n-3).
a(n) ~ sqrt(2 + sqrt(5)*phi^(-1/3) + sqrt(5)*phi^(1/3)) * 3^n * (1 + phi^(-2/3) + phi^(2/3))^n / (2*Pi*n), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio.
(End)

A336293 a(n) = Sum_{k=0..n} binomial(n,k)^2 * binomial(2k,k) (n-k)!.

Original entry on oeis.org

1, 3, 16, 116, 1038, 10922, 131256, 1766592, 26253702, 426173906, 7492270416, 141661870088, 2864168171596, 61621248390756, 1404853103594128, 33815954626749600, 856680253728250950, 22777071459869216850, 633968368216974945600, 18430976777427663053400

Offset: 0

Ilya Gutkovskiy, Jul 16 2020

nonn

Cf. A000984, A002720, A002893.

Table[Sum[Binomial[n, k]^2 Binomial[2 k, k] (n - k)!, {k, 0, n}], {n, 0, 19}]
Table[n! HypergeometricPFQ[{1/2, -n}, {1, 1}, -4], {n, 0, 19}]
nmax = 19; CoefficientList[Series[Exp[x] BesselI[0, 2 Sqrt[x]]^2, {x, 0, nmax}], x] Range[0, nmax]!^2

a(n) = sum(k=0, n, binomial(n,k)^2 * binomial(2*k,k) * (n-k)!); \\ Michel Marcus, Jul 17 2020

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x) * BesselI(0,2*sqrt(x))^2.
a(n) ~ n^n * exp(4*sqrt(n) - n - 2) / sqrt(8*Pi) * (1 + 55/(24*sqrt(n))). - Vaclav Kotesovec, Aug 04 2022
Recurrence: n*a(n) = (3*n^2 + n - 1)*a(n-1) - (n-1)^2*(3*n + 1)*a(n-2) + (n-2)^2*(n-1)^2*a(n-3). - Vaclav Kotesovec, Aug 04 2022

cp's OEIS Frontend

A169711 The function W_n(6) (see Borwein et al. reference for definition).

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A248586 a(n) = Sum_{i=0..n} C(n,i)*C(2i,i)^2.

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A336270 a(n) = Sum_{k=0..n} Sum_{j=0..k} (binomial(n,k) * binomial(k,j))^n.

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A336271 a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k+1) * binomial(n,k)^2 * binomial(2*k,k) * a(n-k).

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A172434 G.f.: Sum_{n>=0} a(n)*x^n/n!^4 = [ Sum_{n>=0} x^n/n!^4 ]^3.

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A180350 G.f.: Sum_{n>=0} a(n)*x^n/n!^5 = [ Sum_{n>=0} x^n/n!^5 ]^3.

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A186377 a(n) equals the least sum of the squares of the coefficients in (1 + 2*x^k + x^p + x^q)^n found at sufficiently large p and q>(n+1)p for some fixed k>0.

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A336638 Sum_{n>=0} a(n) * x^n / (n!)^2 = 1 / BesselJ(0,2*sqrt(x))^3.

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A336271 a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k+1) * binomial(n,k)^2 * binomial(2k,k) a(n-k).

A208426 Expansion of Sum_{n>=0} (3n)!/n!^3 x^(2n)/(1-3x)^(3*n+1).

A336293 a(n) = Sum_{k=0..n} binomial(n,k)^2 * binomial(2k,k) (n-k)!.