A092815 -id:A092815 - cp's OEIS frontend

A218689 Sum_{k=0..n} C(n,k)^6*C(n+k,k)^6.

1, 65, 93313, 795985985, 8178690000001, 93706344780048065, 1453730786373283012225, 26552497154713885161031745, 513912636558068387176582890625, 10769375530849394271690330588432065, 243282405272735566295972089793676717313, 5763401688773271719278313934033057270226625

Offset: 0

Vaclav Kotesovec, Nov 04 2012

nonn

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

Cf. A001850 (p=1), A005259 (p=2), A092813 (p=3), A092814 (p=4), A092815 (p=5).

Table[Sum[Binomial[n,k]^6*Binomial[n+k,k]^6,{k,0,n}],{n,0,20}]

a(n) ~ (1+sqrt(2))^(6*(2n+1))/(2^(17/4)*(Pi*n)^(11/2)*sqrt(3))

Generally, Sum_{k=0..n} C(n,k)^p*C(n+k,k)^p is asymptotic to (1+sqrt(2))^(p*(2*n+1))/(2^(p/2+3/4)*(Pi*n)^(p-1/2)*sqrt(p)) * (1-(2*p-1)/(4*n)+(4*p^2+24*p-19)*sqrt(2)/(96*p*n))

A092813 Schmidt's problem sum for r = 3.

Original entry on oeis.org

1, 9, 433, 36729, 3824001, 450954009, 58160561761, 7989733343097, 1149808762915201, 171540347534028009, 26338900959100106433, 4140153621102790276137, 663592912043903970182289, 108127319237119098011204937, 17868369859451104998973346433, 2989001418301890511076878884729

Offset: 0

Eric W. Weisstein, Mar 06 2004

Apparently, the diagonal of 1/((1 - x - y)*(1 - z - t)*(1 - u - w) - x*y*z*t*u*w). - Peter Bala, Jun 30 2023

Vincenzo Librandi, Table of n, a(n) for n = 0..200
S. Hassani, J.-M. Maillard, and N. Zenine, On the diagonals of rational functions: the minimal number of variables (unabridged version), arXiv:2502.05543 [math-ph], 2025. See p. 11.
Vaclav Kotesovec, Asymptotic of generalized Apery sequences with powers of binomial coefficients, Nov 04 2012.
Eric Weisstein's World of Mathematics, Schmidt's Problem.

Cf. A001850, A005259, A092814, A092815, A218689.

Table[Sum[Binomial[n,k]^3 Binomial[n+k,k]^3,{k,0,n}],{n, 0, 20}] (*Harvey P. Dale, Apr 26 2011 *)

a(n)=sum(k=0,n,binomial(n,k)^3*binomial(n+k,k)^3); \\ Joerg Arndt, May 11 2013

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

a(n) ~ (1+sqrt(2))^(3*(2*n+1))/(2^(9/4)*(Pi*n)^(5/2)*sqrt(3)). - Vaclav Kotesovec, Nov 04 2012

Prepended missing a(0)=1, Joerg Arndt, May 11 2013

A092814 Schmidt's problem sum for r = 4.

Original entry on oeis.org

1, 17, 2593, 990737, 473940001, 261852948017, 166225611652129, 115586046457265681, 85467827222155042849, 66421846251482628852017, 53755021948680412765238593, 44947131400352317819689905201, 38613445585740736549461528111649, 33942058336306457714420306982430001

Offset: 0

Eric W. Weisstein, Mar 06 2004

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
Eric Weisstein's World of Mathematics, Schmidt's Problem

Cf. A001850, A005259, A092813, A092815, A218689.

Table[Sum[Binomial[n, k]^4 Binomial[n+k, k]^4, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 04 2012 *)
a[k_]:= HypergeometricPFQ[{-k,-k,-k,-k,1+k,1+k,1+k,1+k}, {1,1,1,1,1,1,1}, 1]
Table[ a[k], {k, 0, 20}] (* Gerry Martens, Sep 26 2022 *)

a(n)=sum(k=0,n,binomial(n,k)^4*binomial(n+k,k)^4); \\ Joerg Arndt, May 11 2013

a(n) = sum(k=0..n, binomial(n,k)^4 * binomial(n+k,k)^4 ).

a(n) ~ (1+sqrt(2))^(4*(2n+1))/(2^(15/4)*(Pi*n)^(7/2)). - Vaclav Kotesovec, Nov 04 2012

Prepended missing a(0)=1, Joerg Arndt, May 11 2013

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

A336873 a(n) = Sum_{k=0..n} (binomial(n+k,k) * binomial(n,k))^n.

Original entry on oeis.org

1, 3, 73, 36729, 473940001, 155741521320033, 1453730786373283012225, 415588116056535702096428038017, 3278068950996636050857475073848209555969, 756475486389705843580676191270930552553654909184513, 5850304627708628483969594929628923064185219454493588333628772353

Offset: 0

Seiichi Manyama, Aug 06 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..37
Vaclav Kotesovec, Plot of a(n)/(binomial(n + n/sqrt(2),n/sqrt(2)) * binomial(n,n/sqrt(2)))^n for n = 1..500

Cf. A001850, A005259, A092813, A092814, A092815, A167010, A218689, A336829, A346200.

[(&+[(Binomial(2*j,j)*Binomial(n+j,n-j))^n: j in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 31 2022

a[n_] := Sum[(Binomial[n+k, k] * Binomial[n, k])^n, {k, 0, n} ]; Array[a, 11, 0] (* Amiram Eldar, Aug 06 2020 *)

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

def A336873(n): return sum((binomial(2*j,j)*binomial(n+j, n-j))^n for j in (0..n))
[A336873(n) for n in (0..20)] # G. C. Greubel, Aug 31 2022

a(n)^(1/n) ~ (1 + sqrt(2))^(2*n + 1) / (Pi*sqrt(2)*n). - Vaclav Kotesovec, Jul 10 2021

A379610 a(n) = Sum_{j=0..n} binomial(2j,j)^4 binomial(n,j)^2 * Sum_{k=0..n} binomial(k+j,k-j)^2 * binomial(2j,n-k) binomial(2*j,k-j).

Original entry on oeis.org

1, 16, 2576, 1383568, 873960976, 615833930816, 526152430612496, 502263183380000576, 507670642053018634768, 542989589742114444434176, 609997002301177503142835776, 710665335270156912049219940096, 853208134042451055649274133396496, 1051685299255545900736463773134099328

Offset: 0

Seiichi Manyama, Dec 27 2024

nonn

Eric Weisstein's World of Mathematics, Schmidt's Problem

Fifth row of array A094424.

Cf. A092815.

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

A092815(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+k,k) * a(k).

cp's OEIS Frontend

A218689 Sum_{k=0..n} C(n,k)^6*C(n+k,k)^6.

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A092813 Schmidt's problem sum for r = 3.

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A092814 Schmidt's problem sum for r = 4.

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

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A336873 a(n) = Sum_{k=0..n} (binomial(n+k,k) * binomial(n,k))^n.

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A379610 a(n) = Sum_{j=0..n} binomial(2*j,j)^4 * binomial(n,j)^2 * Sum_{k=0..n} binomial(k+j,k-j)^2 * binomial(2*j,n-k) * binomial(2*j,k-j).

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A379610 a(n) = Sum_{j=0..n} binomial(2j,j)^4 binomial(n,j)^2 * Sum_{k=0..n} binomial(k+j,k-j)^2 * binomial(2j,n-k) binomial(2*j,k-j).