A218689 -id:A218689 - cp's OEIS frontend

A092813 Schmidt's problem sum for r = 3.

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

A092815 Schmidt's problem sum for r = 5.

Original entry on oeis.org

1, 33, 15553, 27748833, 61371200001, 155741521320033, 487874692844719489, 1730097641006678817249, 6559621957318406477234689, 26511434186466256434467280033, 113203209912753307355868621335553, 503697803885283278416185835107071649, 2318764463485777975432760948801307487809

Offset: 0

Eric W. Weisstein, Mar 06 2004

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

Cf. A001850, A005259, A092813, A092814, A218689.

Table[Sum[Binomial[n, k]^5 Binomial[n+k, k]^5, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 04 2012 *)

a(n)=sum(k=0,n,binomial(n,k)^5*binomial(n+k,k)^5); \\ Joerg Arndt, May 11 2013

a(n) = sum(k=0..n, binomial(n,k)^5 * binomial(n+k,k)^5 ). - corrected by Vaclav Kotesovec, Nov 04 2012

a(n) ~ (1+sqrt(2))^(5*(2n+1))/(2^(13/4)*(Pi*n)^(9/2)*sqrt(5)). - 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

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

Original entry on oeis.org

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

A014180 Sum_{k = 0..n} binomial(n,k)^3*binomial(n+k,k)^2.

Original entry on oeis.org

1, 5, 109, 3533, 133501, 5629505, 254899765, 12129399245, 599084606845, 30455459491505, 1584249399505609, 83970120618566825, 4520585403820052581, 246592348286170615097, 13603606921687170927109, 757808346139996787715533, 42575668004558257371188605, 2410024012619343278147357297

Offset: 0

N. J. A. Sloane

Compare with the Apéry numbers A005258 and A005259.

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. A218693, A112019, A111968, A014178, A218689, A218692, A005258, A005259.

Table[Sum[Binomial[n,k]^3*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)^3*binomial(n+k,k)^2 ); \\ Joerg Arndt, May 04 2013

a(n) ~ (1+r)^(4*n+5/2)/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)^3*(1+r)^2 = r^5, r = 0.65039847669867... - Vaclav Kotesovec, Nov 04 2012

The expansions exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 5*x + 67*x^2 + 1471*x^3 + 41456*x^4 + 1380268*x^5 + ... and exp( Sum_{n >= 1} a(n-1)*x^n/n ) = 1 + x + 3*x^2 + 39*x^3 + 924*x^4 + 27696*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

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

cp's OEIS Frontend

A092813 Schmidt's problem sum for r = 3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A092815 Schmidt's problem sum for r = 5.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A092814 Schmidt's problem sum for r = 4.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A014180 Sum_{k = 0..n} binomial(n,k)^3*binomial(n+k,k)^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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