A037972 -id:A037972 - cp's OEIS frontend

A329444 The sixth moments of the squared binomial coefficients; a(n) = Sum_{m=0..n} m^6*binomial(n, m)^2.

0, 1, 68, 1314, 18080, 197350, 1836792, 15233316, 115776768, 821760390, 5520171800, 35438827996, 219038609088, 1310833221724, 7629754810160, 43348888067400, 241117582878720, 1316197491501510, 7065439665315480, 37362065079691500, 194909773207512000, 1004374157379474420

Offset: 0

Nikita D. Gogin, Nov 16 2019

nonn

H. W. Gould, Combinatorial Identities, 1972. (See formulas 3.77, 3.78, and 3.79 on page 31.)

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A037966, A037972, A074334, A294486, A329521, A329913.

[(&+[Binomial(n,k)^2*k^6: k in [0..n]]): n in [0..30]]; // G. C. Greubel, Jun 23 2022

Table[Sum[m^6*(Binomial[n, m])^2, {m, 0, n}], {n, 21}]

a(n) = sum(m=0, n, m^6*binomial(n, m)^2); \\ Jinyuan Wang, Nov 23 2019

[n^3*(n+1)*(n^6+3*n^5-13*n^4-15*n^3+30*n^2+8*n-2)*catalan_number(n)/(8*(2*n-1)*(2*n-3)*(2*n-5)) for n in (0..30)] # G. C. Greubel, Jun 23 2022

a(n) = binomial(2*n, n) * n^3*(n^6 + 3*n^5 - 13*n^4 - 15*n^3 + 30*n^2 + 8*n - 2)/(8*(2*n-1)*(2*n-3)*(2*n-5)).

G.f.: x*(1 + 42*x - 168*x^2 + 1648*x^3 - 7608*x^4 + 18144*x^5 - 19376*x^6 - 1440*x^7 + 14400*x^8)/((1-4*x)^6*sqrt(1-4*x)). - G. C. Greubel, Jun 23 2022

A074334 a(n) = Sum_{r=1..n} r^4*binomial(n,r)^2.

Original entry on oeis.org

0, 1, 20, 234, 2144, 16750, 117432, 761460, 4654848, 27173718, 152867000, 834212236, 4438175040, 23108423884, 118111709744, 594059985000, 2946077521920, 14429322555750, 69892354873080, 335194270938780, 1593211647720000, 7511501237722020, 35153884344493200

Offset: 0

Paul Boddington, Mar 05 2003

H. W. Gould, Combinatorial Identities, 1972. (See formulas 3.77, 3.78, and 3.79 on page 31.)

Andrew Howroyd, Table of n, a(n) for n = 0..200
Nikita Gogin and Mika Hirvensalo, On the Moments of Squared Binomial Coefficients, (2020).

Cf. A000108 (Catalan numbers).

Cf. A000984, A002457, A037966, A037972, A329444, A329913.

[n le 1 select n else n^2*(n^3+n^2-3*n-1)*Catalan(n-2): n in [0..30]]; // G. C. Greubel, Jun 23 2022

Total/@Table[r^4 Binomial[n,r]^2,{n,0,20},{r,n}] (* Harvey P. Dale, Dec 04 2017 *)
Table[n^2*(n^3+n^2-3*n-1)*CatalanNumber[n-2] -Boole[n==1], {n,0,30}] (* G. C. Greubel, Jun 23 2022 *)

vector(30, n, n--; sum(k=1, n, k^4*binomial(n,k)^2)) \\ Michel Marcus, Aug 19 2015

[n^2*(n^3+n^2-3*n-1)*catalan_number(n-2) for n in (0..30)] # G. C. Greubel, Jun 23 2022

For n>1 a(n) = n^2*(n^3+n^2-3*n-1)*C(n-2). Here C(n-2) = binomial(2*n-4, n-2)/(n-1) is a Catalan number.
From G. C. Greubel, Jun 23 2022: (Start)
a(n) = (n^2*(n^3 + n^2 - 3*n -1)/(2*(2*n-3)))*binomial(2*n-2, n-1).
G.f.: x*(1 + 2*x + 32*x^3 - 128*x^4 + 144*x^5)/(1-4*x)^(9/2).
E.g.f.: x*exp(2*x)*( (1+2*x)*(1 +6*x +4*x^2)*BesselI(0, 2*x) + 2*x*(2 + 7*x + 4*x^2)*BesselI(1, 2*x) ). (End)
D-finite with recurrence (n-1)*(39*n-106)*a(n) +4*(-38*n^2+n+290)*a(n-1) +4*(100*n^2-784*n+1145)*a(n-2) -64*(13*n+4)*(2*n-9)*a(n-3)=0. - R. J. Mathar, Sep 13 2024

Terms a(18) and beyond from Andrew Howroyd, Jan 16 2020

A329913 The fifth moments of the squared binomial coefficients; a(n) = Sum_{m=0..n} m^5*binomial(n, m)^2.

Original entry on oeis.org

0, 1, 36, 540, 6080, 56250, 455112, 3342192, 22809600, 146988270, 904475000, 5358254616, 30750385536, 171773279860, 937514244240, 5014575000000, 26351064760320, 136319273714070, 695429503781400, 3503580441563400, 17452918098000000, 86055711108818220

Offset: 0

Nikita D. Gogin, Nov 24 2019

nonn

H. W. Gould, Combinatorial Identities, 1972. (See formulas 3.77, 3.78, and 3.79 on page 31.)

Robert Israel, Table of n, a(n) for n = 0..1638

Cf. A074334, A294486.

Cf. A000984, A002457, A037966, A037972, A074334, A329444.

[(&+[Binomial(n,k)^2*k^5: k in [0..n]]): n in [0..30]]; // G. C. Greubel, Jun 23 2022

seq( binomial(2*n,n)*n^4*(n^3 + 3*n^2 - 3*n - 5)/((16*n-8)*(2*n-3)),n=0..30); # Robert Israel, Jan 26 2020

Table[Sum[m^5*(Binomial[n, m])^2, {m, 0, n}], {n, 21}]

a(n) = sum(k=0, n, k^5*binomial(n, k)^2); \\ Michel Marcus, Nov 24 2019

[n^4*(n+1)*(n^3+3*n^2-3*n-5)/(8*(2*n-1)*(2*n-3))*catalan_number(n) for n in (0..30)] # G. C. Greubel, Jun 23 2022

a(n) = binomial(2*n,n)*n^4*(n^3 + 3*n^2 - 3*n - 5)/(8*(2*n-1)*(2*n-3)).
G.f.: x*(1 + 14*x - 54*x^2 + 404*x^3 - 1544*x^4 + 2880*x^5 - 2160*x^6)/(1-4*x)^(11/2). - Stefano Spezia, Jan 03 2020
(-12960 + 8640*n)*a(n) + (7200 - 13680*n)*a(n + 1) + (3920 + 9056*n)*a(n + 2) + (-4184 - 3160*n)*a(n + 3) + (1404 + 620*n)*a(n + 4) + (-584 - 110*n)*a(n + 5) + (14 + 10*n)*a(n + 6) + (n + 6)*a(n + 7) = 0. - Robert Israel, Jan 26 2020

A336828 a(n) = Sum_{k=0..n} binomial(n,k)^2 * k^n.

Original entry on oeis.org

1, 1, 8, 108, 2144, 56250, 1836792, 71799504, 3269445888, 169974711630, 9934458411800, 644825382429096, 46022332032100800, 3582265183110626740, 302002255041807372080, 27413749834141448520000, 2665789990569658618398720, 276477318687585566522176470

Offset: 0

Ilya Gutkovskiy, Aug 05 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..343

Cf. A000984, A002457, A037966, A037972, A072034, A074334, A187021, A329444, A329913, A336214, A341815.

Join[{1}, Table[Sum[Binomial[n, k]^2 k^n, {k, 0, n}], {n, 1, 17}]]

a(n) = sum(k=0, n, binomial(n, k)^2*k^n); \\ Michel Marcus, Aug 05 2020

a(n) ~ c * d^n * (n-1)!, where d = (1 + 2*LambertW(exp(-1/2)/2)) / (4*LambertW(exp(-1/2)/2)^2) = 6.476217542109791521947605963458797355564... and c = 0.21617818094152997942246965143216887599763501682724844713834495... - Vaclav Kotesovec, Feb 20 2021

A037976 a(n) = (1/4)(binomial(4n, 2n) - (-1)^nbinomial(2n, n) + (1-(-1)^n)binomial(2*n, n)^2).

Original entry on oeis.org

0, 4, 16, 436, 3200, 78004, 675808, 15919320, 150266880, 3450748180, 34461586016, 774842070600, 8061900244736, 178065876017176, 1912172640160960, 41596867935469936, 458156035085377536

Offset: 0

N. J. A. Sloane

The right-hand side of a binomial coefficient identity in H. W. Gould, Combinatorial Identities, Morgantown, 1972. (See (3.75) on page 31.)

G. C. Greubel, Table of n, a(n) for n = 0..825

Cf. A000108, A037964, A037972.

[(1/4)*((2*n+1)*Catalan(2*n) -(-1)^n*(n+1)*Catalan(n) +(1-(-1)^n)*(n+1)^2*Catalan(n)^2): n in [0..30]]; // G. C. Greubel, Jun 22 2022

A037976 := proc(n)
    binomial(4*n,2*n)/4-(-1)^n*binomial(2*n,n)/4+(1-(-1)^n)*binomial(2*n,n)^2/4 ;
end proc:
seq(A037976(n),n=0..30) ; # R. J. Mathar, Jul 26 2015

With[{B=Binomial}, Table[(1/4)*(B[4*n,2*n] +B[2*n,n]^2 -2*(-1)^n*B[B[2*n,n] +1, 2]), {n,0,30}]] (* G. C. Greubel, Jun 22 2022 *)

b=binomial; [(1/4)*(b(4*n, 2*n) -(-1)^n*b(2*n, n) +(1-(-1)^n)*b(2*n, n)^2) for n in (0..30)] # G. C. Greubel, Jun 22 2022

From G. C. Greubel, Jun 22 2022: (Start)
a(n) = Sum_{k=0..floor((n-1)/2)} binomial(2*n, 2*k+1)^2.
a(n) = (1/4)*( (2*n+1)*A000108(2*n) - (-1)^n*(n+1)*A000108(n) + (1-(-1)^n)*(n+1)^2*A000108(n)^2 ).
G.f.: (1/4)*(sqrt(1 + sqrt(1-16*x))/(sqrt(2)*sqrt(1-16*x)) - 1/sqrt(1+4*x)) + (1/(2*Pi))*( EllipticK(16*x) - EllipticK(-16*x)). (End)

A336955 a(n) = Sum_{k=0..n} k^k * binomial(n, k)^2.

Original entry on oeis.org

1, 2, 9, 73, 849, 12651, 228493, 4836301, 117204545, 3196763983, 96842596701, 3224356269597, 116981406934417, 4591908332288837, 193846634326107701, 8755364023207809301, 421214258699748184321, 21500563181275847468503, 1160430732790051008442141, 66020998289431649938896445

Offset: 0

Seiichi Manyama, Aug 09 2020

nonn

Cf. A037966, A037972, A086331, A329913, A329444, A336828.

a[n_] := 1 + Sum[k^k * Binomial[n, k]^2, {k, 1, n}]; Array[a, 20, 0] (* Amiram Eldar, Aug 09 2020 *)

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

cp's OEIS Frontend

A329444 The sixth moments of the squared binomial coefficients; a(n) = Sum_{m=0..n} m^6*binomial(n, m)^2.

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A074334 a(n) = Sum_{r=1..n} r^4*binomial(n,r)^2.

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A329913 The fifth moments of the squared binomial coefficients; a(n) = Sum_{m=0..n} m^5*binomial(n, m)^2.

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A336828 a(n) = Sum_{k=0..n} binomial(n,k)^2 * k^n.

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A037976 a(n) = (1/4)*(binomial(4*n, 2*n) - (-1)^n*binomial(2*n, n) + (1-(-1)^n)*binomial(2*n, n)^2).

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A336955 a(n) = Sum_{k=0..n} k^k * binomial(n, k)^2.

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A037976 a(n) = (1/4)(binomial(4n, 2n) - (-1)^nbinomial(2n, n) + (1-(-1)^n)binomial(2*n, n)^2).