A014180 -id:A014180 - cp's OEIS frontend

A111968 a(n) = Sum_{k=0..n} C(n,k)^2*C(n+k,k)^3, where C := binomial.

1, 9, 325, 17577, 1152501, 84505509, 6664647781, 553268669865, 47710914870133, 4236909872278509, 385139801423145825, 35681384898462925125, 3358273513450241419125, 320308335005997679093125, 30900030366269721747776325, 3010365811746267487293617577

Offset: 0

N. J. A. Sloane, Nov 28 2005

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
C. Elsner, On recurrence formulas for sums involving binomial coefficients, Fib. Q., 43,1 (2005), 31-45.
V. Kotesovec, Asymptotic of generalized Apery sequences with powers of binomial coefficients, Nov 04 2012

Cf. A218693, A112019, A014178, A014180, A218689, A218692.

Table[Sum[Binomial[n,k]^2*Binomial[n+k,k]^3,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Nov 04 2012 *)

a(n) = sum(k=0, n, binomial(n, k)^2*binomial(n+k,k)^3); \\ Michel Marcus, Mar 10 2016

a(n) ~ (1+r)^(6*n+7/2)/r^(5*n+9/2)/(4*Pi^2*n^2)*sqrt((1-r)/(5-r)), where r is positive real root of the equation (1-r)^2*(1+r)^3=r^5, r = 0.77591859532439... - Vaclav Kotesovec, Nov 04 2012

Recurrence: (n-1)^2*n^4*(843719*n^6 - 10346391*n^5 + 52472779*n^4 - 140788713*n^3 + 210641238*n^2 - 166531044*n + 54330068)*a(n) = (n-1)^2*(109683470*n^10 - 1564397770*n^9 + 9692963299*n^8 - 34227043418*n^7 + 75994068609*n^6 - 110509975758*n^5 + 106422212572*n^4 - 67092633284*n^3 + 26619112256*n^2 - 6034674112*n + 596279040)*a(n-1) - (1736373702*n^12 - 31711114890*n^11 + 261518988565*n^10 - 1286766506127*n^9 + 4203065855621*n^8 - 9590033857111*n^7 + 15649936441072*n^6 - 18370855225904*n^5 + 15360506258964*n^4 - 8896962441876*n^3 + 3377234408016*n^2 - 751582555104*n + 73915071552)*a(n-2) - (n-2)^2*(36279917*n^10 - 590014481*n^9 + 4216923435*n^8 - 17398379754*n^7 + 45760527058*n^6 - 79915647314*n^5 + 93501944898*n^4 - 72055169071*n^3 + 34824212688*n^2 - 9481092472*n + 1100757336)*a(n-3) - (n-2)^2*(843719*n^6 - 5284077*n^5 + 13396609*n^4 - 17487127*n^3 + 12303648*n^2 - 4393232*n + 621656)*(n-3)^4*a(n-4). - Vaclav Kotesovec, Nov 04 2012

A112019 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+k,k)^2.

Original entry on oeis.org

1, 5, 55, 749, 11251, 178835, 2949115, 49906925, 860905315, 15071939255, 266982872905, 4774722189275, 86070844191775, 1561948324845095, 28507384046515555, 522867506128197869, 9631571375362268515, 178094411589895650815, 3304192479145474141741, 61487420580006795749999

Offset: 0

N. J. A. Sloane, Nov 28 2005

nonn

Diagonal of rational function 1/(1 - x - y - z - x*y + x*z + x*y*z). - Gheorghe Coserea, Jul 01 2018

Diagonal of the rational function 1 / ((1-x)*(1-y)*(1-z) - z). - Seiichi Manyama, Apr 30 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..200
C. Elsner, On recurrence formulas for sums involving binomial coefficients, Fib. Q., 43,1 (2005), 31-45.
V. Kotesovec, Asymptotic of generalized Apery sequences with powers of binomial coefficients, Nov 04 2012

Cf. A218693, A111968, A014178, A014180, A218689, A218692.

Cf. A383536, A383537, A383538.

seq(add((multinomial(n+k,n-k,k,k))*binomial(n+k,k),k=0..n),n=0..19); # Zerinvary Lajos, Oct 18 2006
ogf := hypergeom([1/12,5/12],[1], -1728*(x^3+5*x^2+39*x-2)*x^4 / (x^4+4*x^3+30*x^2-20*x+1)^3 ) / (x^4+4*x^3+30*x^2-20*x+1)^(1/4);
series(ogf, x=0, 30); # Mark van Hoeij, Jan 22 2013

Table[HypergeometricPFQ[{-n, 1 + n, 1 + n}, {1, 1}, -1], {n, 0, 20}] (* Olivier Gérard, Apr 23 2009 *)
Table[Sum[Binomial[n,k]*Binomial[n+k,k]^2,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Nov 04 2012 *)

a(n) = sum(k=0, n, binomial(n,k)*binomial(n+k,k)^2); \\ Michel Marcus, Mar 09 2016

a(n) = 3F2( {-n, 1 + n, 1 + n} ; {1, 1} )(-1). - Olivier Gérard, Apr 23 2009
a(n) ~ (1+r)^(4*n+5/2)/r^(3*n+5/2)/(2*Pi*n)*sqrt((1-r)/(3-r)), where r is positive real root of the equation (1-r)*(1+r)^2=r^3, r = 1/6*((44-3*sqrt(177))^(1/3)+(44+3*sqrt(177))^(1/3)-1) = 0.82948354095849... - Vaclav Kotesovec, Nov 04 2012
Recurrence: 2*n^2*(59*n - 83)*a(n) = (2301*n^3 - 5538*n^2 + 3797*n - 800)*a(n-1) + 5*(59*n^3 - 201*n^2 + 213*n - 64)*a(n-2) + (59*n - 24)*(n-2)^2*a(n-3). - Vaclav Kotesovec, Nov 04 2012
G.f. y=A(x) satisfies: 0 = x*(5*x + 8)*(x^3 + 5*x^2 + 39*x - 2)*y'' + (15*x^4 + 82*x^3 + 315*x^2 + 624*x - 16)*y' + (5*x^3 + 21*x^2 + 80)*y. - Gheorghe Coserea, Jul 01 2018

A014178 a(n) = Sum_{k = 0..n} binomial(n,k)^3*binomial(n+k,k).

Original entry on oeis.org

1, 3, 31, 399, 5871, 93753, 1577479, 27556623, 495001327, 9085988613, 169675769781, 3213444254133, 61573700137431, 1191526503165729, 23252920338835911, 457112339182896399, 9043566887755775727, 179928068420530483389, 3597714616543167088921

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
C. Elsner, On recurrence formulas for sums involving binomial coefficients, Fib. Q., 43,1 (2005), 31-45.
V. Kotesovec, Asymptotic of generalized Apery sequences with powers of binomial coefficients, Nov 04 2012

Cf. A218693, A112019, A111968, A014180, A218689, A218692, A005258, A005259.

f := n->sum( 'binomial(n,k)^3*binomial(n+k,k)^1','k'=0..n);

Table[Sum[Binomial[n,k]^3*Binomial[n+k,k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Nov 04 2012 *)

a(n)=sum(k=0,n, binomial(n,k)^3*binomial(n+k,k) ); \\ Joerg Arndt, May 04 2013

a(n) ~ (1+r)^(2*n+3/2)/r^(4*n+7/2)/(4*Pi^(3/2)*n^(3/2))*sqrt((1-r)/(2+r)), where r is positive real root of the equation (1-r)^3*(1+r)=r^4, r = 0.58252220781047... - Vaclav Kotesovec, Nov 04 2012
Recurrence: (n-1)*n^3*(29412*n^4 - 246240*n^3 + 764259*n^2 - 1042332*n + 527381)*a(n) = 2*(n-1)*(235296*n^7 - 2322864*n^6 + 9245766*n^5 - 19022421*n^4 + 21621181*n^3 - 13561627*n^2 + 4459053*n - 605664)*a(n-1) + 2*(1647072*n^8 - 20377728*n^7 + 107506956*n^6 - 315721020*n^5 + 564159163*n^4 - 627527310*n^3 + 423779896*n^2 - 158592459*n + 25128864)*a(n-2) + 2*(n-2)*(4176504*n^7 - 49583844*n^6 + 243933522*n^5 - 641841009*n^4 + 971188553*n^3 - 841622632*n^2 + 385567572*n - 72023040)*a(n-3) - 4*(n-2)*(29412*n^4 - 128592*n^3 + 202011*n^2 - 134886*n + 32480)*(n-3)^3*a(n-4). - Vaclav Kotesovec, Nov 04 2012
The expansions exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 3*x + 20*x^2 + 184*x^3 + 2060*x^4 + 26246*x^5 + ... and exp( Sum_{n >= 1} a(n-1)*x^n/n ) = 1 + x + 2*x^2 + 12*x^3 + 112*x^4 + 1296*x^5 + ... appear to have integer coefficients. Cf. A005258 and A005259. - Peter Bala, Jan 14 2016

A218693 a(n) = Sum_{k=0..n} C(n,k)*C(n+k,k)^3.

Original entry on oeis.org

1, 9, 271, 11193, 535251, 27854739, 1531656211, 87547358649, 5149886133907, 309721191497259, 18957806551405701, 1177134132932168739, 73964787438524189871, 4694347514292389411103, 300499277330710307643771, 19378727805024033594228153, 1257802636907811605342461587

Offset: 0

Vaclav Kotesovec, Nov 04 2012

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Vaclav Kotesovec, Asymptotic of generalized Apery sequences with powers of binomial coefficients, Nov 04 2012

Cf.
A001850 (p=1, q=1),
A112019 (p=1, q=2),
A005258 (p=2, q=1),
A005259 (p=2, q=2),
A111968 (p=2, q=3),
A014178 (p=3, q=1),
A014180 (p=3, q=2),
A092813 (p=3, q=3),
A218690 (p=4, q=2),
A092814 (p=4, q=4),
A092815 (p=5, q=5),
A218692 (p=6, q=3),
A218689 (p=6, q=6).

Table[Sum[Binomial[n,k]*Binomial[n+k,k]^3,{k,0,n}],{n,0,20}]
a[n_] := HypergeometricPFQ[{-n, n+1, n+1 ,n+1},{1, 1, 1}, -1]; Table[a[n],{n,0,16}] (* Detlef Meya, May 25 2024 *)

Recurrence: 4*(n-1)*n^3*(29412*n^4 - 224352*n^3 + 632931*n^2 - 781692*n + 356309)*a(n) = 2*(n-1)*(4176504*n^7 - 38122740*n^6 + 140783586*n^5 - 270139161*n^4 + 288226505*n^3 - 170040251*n^2 + 51625509*n - 6283008)*a(n-1) + 2*(1647072*n^8 - 19152000*n^7 + 94636812*n^6 - 258386460*n^5 + 423728203*n^4 - 423743982*n^3 + 249392728*n^2 - 77793627*n + 9736704)*a(n-2) + 2*(n-2)*(235296*n^7 - 2618352*n^6 + 11905158*n^5 - 28432149*n^4 + 38188669*n^3 - 28610816*n^2 + 10954716*n - 1618272)*a(n-3) - (n-2)*(29412*n^4 - 106704*n^3 + 136347*n^2 - 71238*n + 12608)*(n-3)^3*a(n-4).
a(n) ~ (1+r)^(6*n+7/2)/r^(4*n+7/2)/(4*Pi^(3/2)*n^(3/2))*sqrt((1-r)/(2-r)), where r is positive real root of the equation (1-r)*(1+r)^3=r^4, r = 0.90340819201887...
Generally, Sum_{k=0..n} C(n,k)^p*C(n+k,k)^q is asymptotic to sqrt((r*(1-r^2))/((p+q+(p-q)*r)*(2*Pi*n)^(p+q-1))) * (1+r)^(q*n)/(1-r)^(p*n+p), where r is positive real root of the equation (1-r)^p*(1+r)^q=r^(p+q). - Vaclav Kotesovec, Nov 07 2012
a(n) = hypergeom([-n, n+1, n+1, n+1],[1, 1, 1], -1). - Detlef Meya, May 25 2024

cp's OEIS Frontend

A111968 a(n) = Sum_{k=0..n} C(n,k)^2*C(n+k,k)^3, where C := binomial.

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A112019 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+k,k)^2.

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A014178 a(n) = Sum_{k = 0..n} binomial(n,k)^3*binomial(n+k,k).

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A218693 a(n) = Sum_{k=0..n} C(n,k)*C(n+k,k)^3.

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