A166337 -id:A166337 - cp's OEIS frontend

A268148 A double binomial sum involving absolute values.

0, 8, 768, 30720, 917504, 23592960, 553648128, 12213813248, 257698037760, 5257039970304, 104453604638720, 2031897488130048, 38843546786070528, 731834939447705600, 13618885273168379904, 250760427251989217280, 4574792530279968800768, 82788987402808467652608

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 (48,-768,4096).

Cf. A000984, A002894, A166337, A254408.

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)^2));

concat(0, Vec(8*x*(1+48*x)/(1-16*x)^3 + 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)^2).
From Colin Barker, Feb 11 2016: (Start)
a(n) = 2^(4*n-1)*n*(2*n-1).
a(n) = 48*a(n-1)-768*a(n-2)+4096*a(n-3) for n>2.
G.f.: 8*x*(1+48*x) / (1-16*x)^3.
(End)

A268147 A double binomial sum involving absolute values.

Original entry on oeis.org

0, 16, 512, 12288, 262144, 5242880, 100663296, 1879048192, 34359738368, 618475290624, 10995116277760, 193514046488576, 3377699720527872, 58546795155816448, 1008806316530991104, 17293822569102704640, 295147905179352825856, 5017514388048998039552

Offset: 0

Richard P. Brent, Jan 27 2016

A fast algorithm follows from Theorem 1 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 (32,-256).

Cf. A000984, A002894, A166337.

a:= proc(n) option remember;
      16*`if`(n<2, n, n*a(n-1)/(n-1))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Jan 29 2016

Table[n*16^n, {n, 0, 20}] (* Jean-François Alcover, Oct 24 2016 *)
LinearRecurrence[{32,-256},{0,16},20] (* Harvey P. Dale, Jul 19 2018 *)

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

concat(0, Vec(16*x/(1-16*x)^2 + O(x^20))) \\ Colin Barker, Feb 11 2016

a(n)=n*16^n \\ Charles R Greathouse IV, May 10 2016

a(n) = Sum_{k=-n..n} (Sum_{l=-n..n} binomial(2*n, n+k)*binomial(2*n, n+l)*abs(k-l)^2).
From Colin Barker, Feb 11 2016: (Start)
a(n) = n*16^n.
a(n) = 32*a(n-1)-256*a(n-2) for n>1.
G.f.: 16*x / (1-16*x)^2.
(End)

A268150 A double binomial sum involving absolute values.

Original entry on oeis.org

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)

A166335 Exponential Riordan array [1+x*sinh(x), x].

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 0, 6, 0, 1, 4, 0, 12, 0, 1, 0, 20, 0, 20, 0, 1, 6, 0, 60, 0, 30, 0, 1, 0, 42, 0, 140, 0, 42, 0, 1, 8, 0, 168, 0, 280, 0, 56, 0, 1, 0, 72, 0, 504, 0, 504, 0, 72, 0, 1, 10, 0, 360, 0, 1260, 0, 840, 0, 90, 0, 1

Offset: 0

Paul Barry, Oct 12 2009

			Triangle begins
1,
0, 1,
2, 0, 1,
0, 6, 0, 1,
4, 0, 12, 0, 1,
0, 20, 0, 20, 0, 1,
6, 0, 60, 0, 30, 0, 1,
0, 42, 0, 140, 0, 42, 0, 1,
8, 0, 168, 0, 280, 0, 56, 0, 1,
0, 72, 0, 504, 0, 504, 0, 72, 0, 1,
10, 0, 360, 0, 1260, 0, 840, 0, 90, 0, 1

Row sums are A131056. Diagonal sums are A166336. Central coefficients are A166337.

(* The function RiordanArray is defined in A256893. *)
RiordanArray[1 + # Sinh[#]&, #&, 11, True] // Flatten (* Jean-François Alcover, Jul 19 2019 *)

Number triangle T(n,k)=[k<=n]*C(n,k)*((n-k)+0^((n-k)/2))(1+(-1)^(n-k))/2.

A268149 A double binomial sum involving absolute values.

Original entry on oeis.org

0, 24, 1120, 33264, 823680, 18475600, 389398464, 7862853600, 153876579840, 2940343837200, 55138611528000, 1018383898440480, 18574619721465600, 335240928272918304, 5996573430996184960, 106438123408375281600, 1876607120325212706816, 32891715945378106711440

Offset: 0

Richard P. Brent, Jan 27 2016

A fast algorithm follows from Theorem 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, A268147.

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

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

Conjecture D-finite with recurrence (2*n-1)*(n-1)*a(n) +2*(-22*n^2+27*n-36)*a(n-1) +12*(4*n-5)*(4*n-7)*a(n-2)=0. - R. J. Mathar, Feb 27 2023

A268151 A double binomial sum involving absolute values.

Original entry on oeis.org

0, 40, 2816, 104448, 3014656, 76021760, 1761607680, 38520487936, 807453851648, 16389595201536, 324355930193920, 6289206510878720, 119908340078739456, 2254051613498933248, 41865462136036130816, 769575104325070356480, 14019525496019259228160, 253384476596474400997376

Offset: 0

Richard P. Brent, Jan 27 2016

A fast algorithm follows from Theorem 1 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 (48,-768,4096).

Cf. A000984, A002894, A166337, A268147, A268149.

LinearRecurrence[{48,-768,4096},{0,40,2816},20] (* Harvey P. Dale, Apr 28 2022 *)

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

concat(0, Vec(8*x*(5+112*x)/(1-16*x)^3 + 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-l)^4).
From Colin Barker, Feb 11 2016: (Start)
a(n) = 2^(4*n-1)*n*(6*n-1).
a(n) = 48*a(n-1)-768*a(n-2)+4096*a(n-3) for n>2.
G.f.: 8*x*(5+112*x) / (1-16*x)^3.
(End)

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A268148 A double binomial sum involving absolute values.

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A268147 A double binomial sum involving absolute values.

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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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A166335 Exponential Riordan array [1+x*sinh(x), x].

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A268149 A double binomial sum involving absolute values.

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A268151 A double binomial sum involving absolute values.

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Author

Keywords

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Crossrefs

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Mathematica

PARI

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Formula