A268148 -id:A268148 - cp's OEIS frontend

A268150 A double binomial sum involving absolute values.

0, 8, 2496, 177120, 7616000, 255780000, 7410154752, 194544814464, 4760448675840, 110493063252000, 2461297261280000, 53051182041906048, 1113060644163127296, 22833886572836393600, 459594580755139200000, 9100826722891800000000, 177680489488222659379200, 3426237501864596491802400

Offset: 0

Richard P. Brent, Jan 27 2016

A fast algorithm follows from Lemma 1 of Brent et al. article.

Richard P. Brent, Hideyuki Ohtsuka, Judy-anne H. Osborn, Helmut Prodinger, Some binomial sums involving absolute values, arXiv:1411.1477v2 [math.CO], 2016.

Cf. A000984, A002894, A166337, A254408, A268148.

A268150 := proc(n)
    add( add( binomial(2*n,n+k)*binomial(2*n,n+l)*abs(k^2-l^2)^3,l=-n..n),k=-n..n) ;
end proc:
seq(A268150(n),n=0..10) ; # R. J. Mathar, Feb 27 2023

a(n) = sum(k=-n,n, sum(l=-n,n, binomial(2*n, n+k)*binomial(2*n, n+l)*abs(k^2 - l^2)^3));

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

Conjecture D-finite with recurrence -(4621*n-8921)*(n-1)^2*a(n) +4*(148256*n^3 -1055204*n^2 +2794799*n -2529792)*a(n-1) -64*(32443*n- 32400)*(2*n-3)*(2*n-5)*a(n-2)=0. - R. J. Mathar, Feb 27 2023

A268152 A double binomial sum involving absolute values.

Original entry on oeis.org

0, 8, 8832, 1228800, 79364096, 3562536960, 129276837888, 4079413624832, 116608362086400, 3096396542509056, 77661255048888320, 1861218099127123968, 42980384518787039232, 962362945373732864000, 20993511648589057622016, 447858123072052742062080, 9371462498278516088373248

Offset: 0

Richard P. Brent, Jan 27 2016

A fast algorithm follows from Theorem 5 of Brent et al. article.

Colin Barker, Table of n, a(n) for n = 0..800
Richard P. Brent, Hideyuki Ohtsuka, Judy-anne H. Osborn, Helmut Prodinger, Some binomial sums involving absolute values, arXiv:1411.1477v2 [math.CO], 2016.
Index entries for linear recurrences with constant coefficients, signature (80,-2560,40960,-327680,1048576).

Cf. A000984, A002894, A166337, A254408, A268148, A268150.

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

concat(0, Vec(8*x*(1+1024*x+67840*x^2+417792*x^3)/(1-16*x)^5 + O(x^20))) \\ Colin Barker, Feb 11 2016

a(n) = Sum_{k=-n..n} (Sum_{l=-n..n} binomial(2*n, n+k)*binomial(2*n, n+l)*abs(k^2 - l^2)^4).
From Colin Barker, Feb 11 2016: (Start)
a(n) = 4^(2*n-1)*n*(36*n^3-84*n^2+67*n-17).
a(n) = 80*a(n-1)-2560*a(n-2)+40960*a(n-3)-327680*a(n-4)+1048576*a(n-5) for n>4.
G.f.: 8*x*(1+1024*x+67840*x^2+417792*x^3) / (1-16*x)^5.
(End)

A269877 a(n) = 2^(4n-3)n*(2n-1)(900n^4-4500n^3+8895n^2-8055n+2764), a closed form for a double binomial sum involving absolute values.

Original entry on oeis.org

0, 8, 121728, 77214720, 12676235264, 1090372239360, 64922717257728, 3052335748087808, 121762580539637760, 4304417014325182464, 138706918527488491520, 4154140250223566389248, 117243264067548833906688, 3150495258536853477785600, 81236017376284183797694464

Offset: 0

Vincenzo Librandi, May 10 2016

A fast algorithm follows from Theorem 5 of Brent et al. article.

Vincenzo Librandi, Table of n, a(n) for n = 0..800
Richard P. Brent, Hideyuki Ohtsuka, Judy-anne H. Osborn, Helmut Prodinger, Some binomial sums involving absolute values, arXiv:1411.1477 [math.CO], 2016, page 11.
Index entries for linear recurrences with constant coefficients, signature (112,-5376,143360,-2293760,22020096,-117440512,268435456).

Cf. A001025, A268148, A268152.

[2^(4*n-3)*n*(2*n-1)*(900*n^4-4500*n^3+8895*n^2-8055*n+2764): n in [0..20]];

Table[2^(4 n - 3) n (2 n - 1) (900 n^4 - 4500 n^3 + 8895 n^2 - 8055 n + 2764), {n, 0, 15}]
LinearRecurrence[{112,-5376,143360,-2293760,22020096,-117440512,268435456},{0,8,121728,77214720,12676235264,1090372239360,64922717257728},20] (* Harvey P. Dale, Oct 28 2023 *)

G.f.: 8*x*(1 + 15104*x + 7953024*x^2 + 585181184*x^3 + 8538456064*x^4 + 19750453248*x^5)/(1-16*x)^7.
a(n) = Sum_{k=-n..n} (Sum_{l=-n..n} binomial(2*n, n+k)*binomial(2*n, n+l)*(k^2 - l^2)^6).
a(n) = 2^(4*n-3)*n*(2*n-1)*(900*n^4-4500*n^3+8895*n^2-8055*n+2764).

A272913 a(n) = 32^(2n-1)(n-1)n^3binomial(2n,n)^2, a closed form for a triple binomial sum involving absolute values.

Original entry on oeis.org

0, 0, 6912, 2073600, 361267200, 48771072000, 5665247723520, 595732271726592, 58357447026278400, 5420989989833932800, 483204292920999936000, 41671538221507034480640, 3497929581885972295974912, 287077554068924493987840000, 23115688495680026711162880000

Offset: 0

Bruno Berselli, May 10 2016

See Theorem 6 of Brent et al. article.

a(n) is divisible by 48^2.

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Richard P. Brent, Hideyuki Ohtsuka, Judy-anne H. Osborn, Helmut Prodinger, Some binomial sums involving absolute values, arXiv:1411.1477 [math.CO], 2016, page 11.

Cf. A254408, A268147, A268148, A268149, A268150, A268151, A268152, A269877.

[3*2^(2*n-1)*(n-1)*n^3*Binomial(2*n,n)^2: n in [0..20]];

Table[3 2^(2 n - 1) (n - 1) n^3 Binomial[2 n, n]^2, {n, 0, 20}]

vector(20, n, n--; 3*2^(2*n-1)*(n-1)*n^3*binomial(2*n,n)^2)

[3*2^(2*n-1)*(n-1)*n^3*binomial(2*n, n)^2 for n in range(20)]

a(n) = Sum_{i=-n..n} (Sum_{j=-n..n} (Sum_{k=-n..n} binomial(2*n, n+i)*binomial(2*n, n+j)*binomial(2*n, n+k)*|(i^2-j^2)*(i^2-k^2)*(j^2-k^2)|)).
G.f.: 6912*x^2*(2F1(5/2, 5/2, 2, 64*x) + 100*x*2F1(7/2, 7/2, 3, 64*x)), where 2F1() is the Gauss hypergeometric function.
D-finite with recurrence (n-2)*(n-1)^2*a(n) -16*n*(2*n-1)^2*a(n-1)=0. - R. J. Mathar, Feb 08 2021

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A268150 A double binomial sum involving absolute values.

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A268152 A double binomial sum involving absolute values.

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Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

Formula

A269877 a(n) = 2^(4*n-3)*n*(2*n-1)*(900*n^4-4500*n^3+8895*n^2-8055*n+2764), a closed form for a double binomial sum involving absolute values.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A272913 a(n) = 3*2^(2*n-1)*(n-1)*n^3*binomial(2*n,n)^2, a closed form for a triple binomial sum involving absolute values.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A269877 a(n) = 2^(4n-3)n*(2n-1)(900n^4-4500n^3+8895n^2-8055n+2764), a closed form for a double binomial sum involving absolute values.

A272913 a(n) = 32^(2n-1)(n-1)n^3binomial(2n,n)^2, a closed form for a triple binomial sum involving absolute values.