A294486 -id:A294486 - 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

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

A145436 Decimal expansion of Sum_{n>=0} (-1)^n/((2n+1)^2*binomial(2n,n)).

Original entry on oeis.org

9, 5, 0, 2, 3, 9, 6, 0, 5, 1, 1, 6, 6, 4, 3, 2, 5, 8, 9, 8, 1, 6, 2, 7, 9, 5, 2, 9, 5, 1, 4, 2, 6, 9, 0, 9, 1, 6, 9, 7, 3, 0, 8, 5, 1, 0, 5, 8, 9, 0, 1, 8, 2, 5, 2, 8, 9, 6, 5, 4, 5, 4, 3, 3, 0, 0, 6, 2, 1, 4, 3, 3, 7, 0, 2, 3, 1, 5, 4, 3, 4, 8, 7, 8, 4, 6, 5, 2, 5, 9, 3, 6, 0

Offset: 0

R. J. Mathar, Feb 08 2009

			0.95023960511664325898162795295142690916973085105890...

Alexander Apelblat, Tables of Integrals and Series, Harri Deutsch, (1996), 4.1.45.
Bruce C. Berndt, Ramanujan's Notebooks, Part 1, Springer-Verlag, 1985, Chapter 9, p. 289, eq. (vii).
Calvin C. Clawson, Mathematical Mysteries: The Beauty and Magic of Numbers, Springer, 2013. See p. 222.

Cf. A000984, A002390, A013661, A294486.

Maple
```
1/6*Pi^2-3*ln(1/2+1/2*5^(1/2))^2 ;
```

RealDigits[Zeta[2] - 3 * Log[GoldenRatio]^2, 10, 120][[1]] (* Amiram Eldar, May 06 2023 *)

Pi^2/6-3*log(1/2+sqrt(5)/2)^2 \\ Charles R Greathouse IV, Sep 13 2013

Equals A013661 - 3*A002390^2.

A329521 The sixth moments of the alternated squared binomial coefficients; a(n) = Sum_{m=0..n} (-1)^mm^6binomial(n, m)^2.

Original entry on oeis.org

0, -1, 60, -162, -5280, 20250, 128520, -569380, -1854720, 9338490, 20097000, -113704668, -181621440, 1142905764, 1447926480, -10042461000, -10529925120, 79859881530, 71384175720, -587933314540, -457825368000, 4070529226764

Offset: 0

Nikita D. Gogin, Nov 15 2019

sign

H. W. Gould, Combinatorial Identities, 1972.

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

Cf. A074334, A294486, A329444.

[&+[(-1)^m*m^6*(Binomial(n,m))^2:m in [0..n]]:n in [0..21]]; // Marius A. Burtea, Nov 15 2019

Table[Sum[(-1)^m*m^6*(Binomial[n, m])^2, {m, 0, n}], {n, 21}]

a(n) = sum(m=0, n, (-1)^m*m^6*binomial(n , m)^2); \\ Michel Marcus, Nov 15 2019

a(n) = (-1)^((n+2)/2)*binomial(n, n/2)*(n^3*(n+1)*(3n-1)/4), if n is even,
a(n) = (-1)^((n-1)/2)*binomial(n,((n+1)/2))*(n^2*( n+1)*(n^3+n^2-9n+3)/8), if n is odd.
G.f.: x*(-1 + 60*x - 188*x^2 - 3720*x^3 + 15752*x^4 + 8400*x^5 - 90928*x^6 + 79680*x^7 + 42112*x^8 - 69120*x^9 + 17408*x^10)/(1+4*x^2)^(13/2). - Stefano Spezia, Nov 15 2019

A361719 a(n) = Sum_{k = 1..n} (-1)^(n+k) * k^3 * binomial(n,k)^2.

Original entry on oeis.org

0, 1, 4, -36, -96, 450, 1080, -3920, -8960, 28350, 63000, -182952, -399168, 1093092, 2354352, -6177600, -13178880, 33474870, 70887960, -175518200, -369512000, 896251356, 1877859984, -4478082336, -9345563136, 21971267500, 45700236400, -106148523600, -220159900800

Offset: 0

Peter Bala, Mar 24 2023

Compare with the alternating binomial sum evaluation Sum_{k = 0..2*n+1} (-1)^(n+k+1) * k^2 * binomial(2*n+1,k)^2. = (2*n+1)^2 * binomial(2*n,n) = A294486(n).

Winston de Greef, Table of n, a(n) for n = 0..3271
Wikipedia, Apéry's constant

Cf. A000984, A001405, A002117, A294486.

seq(add( (-1)^(n+k)*k^3*binomial(n,k)^2, k = 0..n ), n = 0..20);

a(n) = (-1)^((n-1)*(n+2)/2) * n*((n+1)\2)^2 * binomial(n, n\2) \\ Winston de Greef, Mar 24 2023

from math import comb
def A361719(n): return (-((m:=n>>1)+1)*n**2*comb(n-1,m) if n&2 else ((m:=n>>1)+1)*n**2*comb(n-1,m)) if n&1 else (((m:=n>>1)**3<<1)*comb(n,m) if n&2 else -((m:=n>>1)**3<<1)*comb(n,m)) # Chai Wah Wu, Mar 24 2023

a(n) = (-1)^((n-1)*(n+2)/2) * n*floor((n+1)/2)^2 * binomial(n, floor(n/2)) = (-1)^((n-1)*(n+2)/2) * n*floor((n+1)/2)^2 * A001405(n).
a(2*n) = (-1)^(n+1) * 2*n^3 * binomial(2*n,n).
a(2*n+1) = (-1)^n * (n+1)*(2*n+1)^2 * binomial(2*n,n).
a(n) = (-1)^(n+1) * n^2 * hypergeom([2, 1 - n, 1 - n], [1, 1], -1).
P-recursive: (2*n^2 - 5*n + 4)*(n - 2)*(n - 1)^3*a(n) = 2*n^2*(3*n - 5)*(n - 2)*a(n-1) - 4*n^2*(2*n^2 - n + 1)*(n - 1)^2*a(n-2) with a(0) = 0 and a(1) = 1.
5*Sum_{n >= 1} 1/a(2*n) = zeta(3), a result due to Markov (1890), rediscovered by Apéry (1979). - Peter Bala, Oct 24 2023

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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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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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A145436 Decimal expansion of Sum_{n>=0} (-1)^n/((2n+1)^2*binomial(2n,n)).

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A329521 The sixth moments of the alternated squared binomial coefficients; a(n) = Sum_{m=0..n} (-1)^m*m^6*binomial(n, m)^2.

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A361719 a(n) = Sum_{k = 1..n} (-1)^(n+k) * k^3 * binomial(n,k)^2.

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A329521 The sixth moments of the alternated squared binomial coefficients; a(n) = Sum_{m=0..n} (-1)^mm^6binomial(n, m)^2.