cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A206294 Riordan array (1, x/(1-x)^3).

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 6, 6, 1, 0, 10, 21, 9, 1, 0, 15, 56, 45, 12, 1, 0, 21, 126, 165, 78, 15, 1, 0, 28, 252, 495, 364, 120, 18, 1, 0, 36, 462, 1287, 1365, 680, 171, 21, 1, 0, 45, 792, 3003, 4368, 3060, 1140, 231, 24, 1
Offset: 0

Views

Author

Philippe Deléham, Feb 05 2012

Keywords

Comments

The convolution triangle of the triangular numbers A000217. - Peter Luschny, Oct 07 2022

Examples

			Triangle begins:
1
0, 1
0, 3, 1
0, 6, 6, 1
0, 10, 21, 9, 1
0, 15, 56, 45, 12, 1
0, 21, 126, 165, 78, 15, 1
0, 28, 252, 495, 364, 120, 18, 1
0, 36, 462, 1287, 1365, 680, 171, 21, 1
0, 45, 792, 3003, 4368, 3060, 1140, 231, 24, 1
0, 55, 1287, 6435, 12376, 11628, 5985, 1771, 300, 27, 1
0, 66, 2002, 12870, 31824, 38760, 26324, 10626, 2600, 378, 30, 1

Links

Crossrefs

Cf. Columns: A000007, A000217 (triangular numbers), A000389, A000581, A001288, A010967..(+3)..A011000, A017714..(+3)..A017762.
Row sums are A052529.
Cf. A127893.

Programs

  • Maple
    # Uses function PMatrix from A357368.
PMatrix(10, n -> n * (n + 1) / 2); # Peter Luschny, Oct 07 2022
  • Mathematica
    Table[If[n == 0 && k == 0 , 1, Binomial[n - 1 + 2 k, n - k]], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 25 2017 *)
  • PARI
    {T(n,k)=polcoeff(1/(1-x+x*O(x^(n-k)))^(3*k),n-k)}
  • PARI
    {T(n,k)=polcoeff(polcoeff((1-x)^3/((1-x)^3-y*x +x*O(x^n)),n,x),k,y)}
for(n=0,12,for(k=0,n,print1(T(n,k),", "));print(""))

Formula

Triangle T(n,k), read by rows, given by (0, 3, -1, 2/3, -1/6, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
T(n,0) = 0^n, T(n,k) = C(n-1+2k, n-k) for k > 0.
T(n,n) = 1, T(k+1,k) = 3*k = A008585(k), T(k+2,k) = A081266(k).
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A052529(n), A052910(n) for x = 0, 1, 2 respectively.
G.f.: (1-x)^3/((1-x)^3-y*x).

A376159 G.f. satisfies A(x) = 1 / ((1-x)^3 - x*A(x)).

Original entry on oeis.org

1, 4, 17, 90, 539, 3451, 23100, 159720, 1131905, 8178326, 60019533, 446166771, 3352530190, 25422458170, 194302002463, 1495223230621, 11575504625874, 90090318248607, 704480581789900, 5532228951823605, 43610427926723780, 344972119634359080, 2737451123900901555
Offset: 0

Views

Author

Seiichi Manyama, Sep 12 2024

Keywords

Crossrefs

Cf. A000108, A006318, A162477.
Cf. A052529, A366184, A376160.
Cf. A360100.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(2/((1-x)^3+sqrt((1-x)^6-4*x)))
  • PARI
    a(n) = sum(k=0, n, binomial(n+5*k+2, n-k)*binomial(2*k, k)/(k+1));

Formula

G.f.: 2 / ((1-x)^3 + sqrt((1-x)^6 - 4*x)).
a(n) = Sum_{k=0..n} binomial(n+5*k+2,n-k) * binomial(2*k,k)/(k+1).

A376160 G.f. satisfies A(x) = 1 / ((1-x)^3 - x*A(x)^3).

Original entry on oeis.org

1, 4, 25, 260, 3205, 42966, 609567, 8999164, 136811781, 2127343669, 33675622992, 540878965522, 8792433396559, 144383416380703, 2391557494237062, 39910530610590312, 670383542665237001, 11325278943044058378, 192301381444863249559, 3280101940070399446926
Offset: 0

Views

Author

Seiichi Manyama, Sep 12 2024

Keywords

Crossrefs

Cf. A002293, A349310, A364630.
Cf. A052529, A366184, A376159.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(n+11*k+2, n-k)*binomial(4*k, k)/(3*k+1));

Formula

a(n) = Sum_{k=0..n} binomial(n+11*k+2,n-k) * binomial(4*k,k)/(3*k+1).

A052603 E.g.f. (1-x)^3/(1-4x+3x^2-x^3).

Original entry on oeis.org

1, 1, 8, 78, 984, 15480, 292320, 6441120, 162207360, 4595512320, 144662112000, 5009199148800, 189221439052800, 7743449813299200, 341258374762905600, 16113703632009984000, 811588993992032256000, 43431603596770701312000
Offset: 0

Views

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

Links

Programs

  • Maple
    spec := [S,{S=Sequence(Prod(Z,Sequence(Z),Sequence(Z),Sequence(Z)))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Formula

E.g.f.: (-1+x)^3/(-1+4*x-3*x^2+x^3)
Recurrence: {a(1)=1, a(0)=1, a(2)=8, (-11*n-6-n^3-6*n^2)*a(n)+(18+3*n^2+15*n)*a(n+1)+(-4*n-12)*a(n+2)+a(n+3)=0, a(3)=78}
Sum(-1/31*(5*_alpha+3*_alpha^2-6)*_alpha^(-1-n), _alpha=RootOf(-1+4*_Z-3*_Z^2+_Z^3))*n!
a(n)=n!*A052529(n). - R. J. Mathar, Jun 03 2022

A350498 Convolution of triangular numbers with every third number of Narayana's Cows sequence.

Original entry on oeis.org

0, 1, 7, 31, 114, 385, 1250, 3987, 12619, 39810, 125425, 394955, 1243433, 3914383, 12322293, 38789576, 122105944, 384377494, 1209981891, 3808901216, 11990036895, 37743426054, 118812495000, 374009739009, 1177344897390, 3706162867858, 11666626518622, 36725362368682, 115607732787126, 363921470561515
Offset: 1

Views

Author

Greg Dresden, Jan 04 2022

Keywords

Comments

This is the convolution of N(3*n-1) with t(n); in other words, a(n) = Sum_{i=1..n} N(3*i-1)*t(n-i) where N(k)=A000930(k) is the k-th number in Narayana's Cows sequence and t(k)=A000217(k) is the k-th triangular number.

Examples

			For n=4, a(4) = N(2)*t(3) + N(5)*t(2) + N(8)*t(1) + N(11)*t(0) = 1*6 + 4*3 + 13*1 + 41*0 = 31, where N(k)=A000930(k) and t(k)=A000217(k).

References

  • G. Dresden and M. Tulskikh, "Convolutions of Sequences with Single-Term Signature Differences", preprint.

Links

Crossrefs

Cf. A000217, A000930.

Programs

  • Mathematica
    CoefficientList[
Series[x/((-1 + x)^3 (-1 + 4 x - 3 x^2 + x^3)), {x, 0, 30}], x]

Formula

a(n) = N(3*n-1) - A000217(n) where N(k)=A000930(k).
G.f.: x^2/((1 - x)^3 * (1 - 4*x + 3*x^2 - x^3)).
a(n) = A052529(n)-A000217(n), n>0. - R. J. Mathar, Aug 17 2022
Previous Showing 21-25 of 25 results.

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