A115291 -id:A115291 - cp's OEIS frontend

A115292 Correlation triangle for (1+x)^3/(1-x).

1, 4, 4, 7, 17, 7, 8, 32, 32, 8, 8, 39, 66, 39, 8, 8, 40, 88, 88, 40, 8, 8, 40, 95, 130, 95, 40, 8, 8, 40, 96, 152, 152, 96, 40, 8, 8, 40, 96, 159, 194, 159, 96, 40, 8, 8, 40, 96, 160, 216, 216, 160, 96, 40, 8, 8, 40, 96, 160, 223, 258, 223, 160, 96, 40, 8

Offset: 0

Paul Barry, Jan 19 2006

Row sums are A115293. Diagonal sums are A115294. T(2n,n) is A115295. T(2n,n)-T(2n,n+1) is 1,10,27,35,35,35,... The transform of (-1)^n by this matrix is 1,0,-3,0,4,0,-4,0,4,...

			Triangle begins
1;
4, 4;
7, 17, 7;
8, 32, 32, 8;
8, 39, 66, 39, 8;
8, 40, 88, 88, 40, 8;
8, 40, 95, 130, 95, 40, 8;
8, 40, 96, 152, 152, 96, 40, 8;

G.f.: (1+x)^3*(1+xy)^3/((1-x)(1-x*y)(1-x^2*y)); Number triangle T(n, k)=sum{j=0..n, [j<=k]*A115291(k-j)*[j<=n-k]*A115291(n-k-j)}.

A115293 Row sums of correlation triangle for (1+x)^3/(1-x).

Original entry on oeis.org

1, 8, 31, 80, 160, 272, 416, 592, 800, 1040, 1312, 1616, 1952, 2320, 2720, 3152, 3616, 4112, 4640, 5200, 5792, 6416, 7072, 7760, 8480, 9232, 10016, 10832, 11680, 12560, 13472, 14416, 15392, 16400, 17440, 18512, 19616, 20752, 21920, 23120, 24352

Offset: 0

Paul Barry, Jan 19 2006

Row sums of number triangle A115292.

If Y_i (i=1,2,3,4,5) are 2-blocks of a (n+5)-set X then a(n-2) is the number of 7-subsets of X intersecting each Y_i (i=1,2,3,4,5). - Milan Janjic, Oct 28 2007

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Milan Janjic, Two Enumerative Functions
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A115291, A115292, A262732.

seq(add(binomial(5,n-k)*binomial(k+2,k), k = 0..n), n = 0..40); # Peter Bala, Sep 26 2021

LinearRecurrence[{3,-3,1},{1,8,31,80,160,272},50] (* Harvey P. Dale, Dec 03 2018 *)

a(n) = sum(k = 0, n, binomial(5,n-k)*binomial(k+2,k)); \\ Michel Marcus, Oct 01 2021

G.f.: A(x) = (1+x)^5/(1-x)^3.
a(n) = Sum_{k = 0..n} Sum_{j = 0..n} [j<=k]*A115291(k-j)*[j<=n-k]*A115291(n-k-j).
From Peter Bala, Sep 26 2021: (Start)
a(n) = Sum_{k = 0..n} binomial(5,n-k)*binomial(k+2,k).
A262732(n) = [x^n] A(x)^n. (End)

A115294 Diagonal sums of correlation triangle for (1+x)^3/(1-x).

Original entry on oeis.org

1, 4, 11, 25, 47, 79, 122, 175, 239, 314, 399, 495, 602, 719, 847, 986, 1135, 1295, 1466, 1647, 1839, 2042, 2255, 2479, 2714, 2959, 3215, 3482, 3759, 4047, 4346, 4655, 4975, 5306, 5647, 5999, 6362, 6735, 7119, 7514, 7919, 8335, 8762, 9199, 9647, 10106

Offset: 0

Paul Barry, Jan 19 2006

Diagonal sums of number triangle A115292.

Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

LinearRecurrence[{2,-1,1,-2,1},{1,4,11,25,47,79,122,175,239},50] (* Harvey P. Dale, Jun 11 2017 *)

G.f.: (1+x)^2*(1+x^2)^3/((1-x)^2*(1-x^3)); a(n)=sum{k=0..floor(n/2), sum{j=0..n-k, [j<=k]*A115291(k-j)*[j<=n-2k]*A115291(n-2k-j)}}.

A122431 Riordan array ((1+x)^3,x).

Original entry on oeis.org

1, 3, 1, 3, 3, 1, 1, 3, 3, 1, 0, 1, 3, 3, 1, 0, 0, 1, 3, 3, 1, 0, 0, 0, 1, 3, 3, 1, 0, 0, 0, 0, 1, 3, 3, 1, 0, 0, 0, 0, 0, 1, 3, 3, 1, 0, 0, 0, 0, 0, 0, 1, 3, 3, 1, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Paul Barry, Sep 04 2006

Row sums are A115291. Diagonal sums are A113311. Inverse is A122432. Sequence array for C(3,n).

			Triangle begins
1,
3,1,
3,3,1,
1,3,3,1,
0,1,3,3,1,
0,0,1,3,3,1,
0,0,0,1,3,3,1,
0,0,0,0,1,3,3,1

Number triangle T(n,k)=C(3,n-k)

A171445 Expansion of g.f. (1+z)^(24)/(1-z).

Original entry on oeis.org

1, 25, 301, 2325, 12951, 55455, 190051, 536155, 1271626, 2579130, 4540386, 7036530, 9740686, 12236830, 14198086, 15505590, 16241061, 16587165, 16721761, 16764265, 16774891, 16776915, 16777191, 16777215, 16777216, 16777216

Offset: 0

Richard Choulet, Dec 09 2009

a(n)=2^(24)=16777216 for n>=24. We observe that this sequence is the transform of A171443 by the iterated T^(16) of T such that: T(u_0,u_1,u_2,u_3,u_4,u_5,...)=(u_0,u_0+u_1,u_1+u_2,u_2+u_3,u_3+u_4,...).

			a(3) = C(25,3)+C(25,3-2) = 2325.

Richard Choulet, Une nouvelle formule de combinatoire?, Mathématique et Pédagogie, 157 (2006), p. 53-60. In French.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A040000, A113311, A115291, A171418, A171440, A171441, A171442, A171443.

m:=25:for n from 0 to 40 do a(n):=sum('binomial(m,n-2*k)',k=0..floor(n/2)): od : seq(a(n),n=0..40);

CoefficientList[Series[(1+x)^24/(1-x),{x,0,30}],x] (* Harvey P. Dale, Jun 11 2019 *)

With m=25, a(n) = Sum_{k=0..floor(n/2)} binomial(m,n-2*k).

A296817 Expansion of 1/Sum_{k>=0} (2k+1)^2x^k.

Original entry on oeis.org

1, -9, 56, -328, 1912, -11144, 64952, -378568, 2206456, -12860168, 74954552, -436867144, 2546248312, -14840622728, 86497488056, -504144305608, 2938368345592, -17126065767944, 99818026262072, -581782091804488, 3390874524564856, -19763465055584648

Offset: 0

Seiichi Manyama, Dec 21 2017

Index entries for linear recurrences with constant coefficients, signature (-6,-1).

Cf. A016754, A115291.

f:= gfun:-rectoproc({a(n) = -6 * a(n-1) - a(n-2), a(0)=1, a(1)=-9, a(2)=56, a(3)=-328},a(n),remember):
map(f, [$0..50]); # Robert Israel, Dec 21 2017

CoefficientList[Series[1/Sum[(2*k+1)^2*x^k, {k, 0, 30}], {x, 0, 30}], x] (* Vaclav Kotesovec, Dec 21 2017 *)
f[n_] := Simplify[ 4*(-1)^n*((Sqrt[2] +1)^(2n -1) - (Sqrt[2] -1)^(2n -1))]; f[0] = 1; f[1] = -9; Array[f, 22, 0] (* or *)
CoefficientList[ Series[-(x^3 -3x^2 +3x -1)/(x^2 +6x +1), {x, 0, 21}], x] (* or *)
Join[{1, -9}, LinearRecurrence[{-6, -1}, {56, -328}, 20]] (* Robert G. Wilson v, Dec 21 2017 *)

N=66; x='x+O('x^N); Vec(1/sum(k=0, N, (2*k+1)^2*x^k))

a(n) = -6 * a(n-1) - a(n-2) for n > 3.
For n > 1, a(n) = 4*(-1)^n * ((sqrt(2)+1)^(2*n-1) - (sqrt(2)-1)^(2*n-1)). - Vaclav Kotesovec, Dec 21 2017
G.f.: (1-x)^3/(1+6*x+x^2). - Robert Israel, Dec 21 2017
a(n) = 8*A002315(n-1), n>1. - R. J. Mathar, Jan 27 2020

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A115292 Correlation triangle for (1+x)^3/(1-x).

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A115293 Row sums of correlation triangle for (1+x)^3/(1-x).

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A115294 Diagonal sums of correlation triangle for (1+x)^3/(1-x).

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A122431 Riordan array ((1+x)^3,x).

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A171445 Expansion of g.f. (1+z)^(24)/(1-z).

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A296817 Expansion of 1/Sum_{k>=0} (2*k+1)^2*x^k.

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A296817 Expansion of 1/Sum_{k>=0} (2k+1)^2x^k.