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.

Showing 1-3 of 3 results.

A159965 Riordan array (1/sqrt(1-4x), (1-2x-(1-3x)c(x))/(x*sqrt(1-4x))), c(x) the g.f. of A000108.

Original entry on oeis.org

1, 2, 1, 6, 5, 1, 20, 21, 8, 1, 70, 84, 45, 11, 1, 252, 330, 220, 78, 14, 1, 924, 1287, 1001, 455, 120, 17, 1, 3432, 5005, 4368, 2380, 816, 171, 20, 1, 12870, 19448, 18564, 11628, 4845, 1330, 231, 23, 1, 48620, 75582, 77520, 54264, 26334, 8855, 2024, 300, 26, 1
Offset: 0

Views

Author

Paul Barry, Apr 28 2009

Keywords

Comments

Product of A007318 and A114422. Product of A007318^2 and A116382. Row sums are A108080.
Diagonal sums are A108081.
Riordan array (1/sqrt(1 - 4*x), x*c(x)^3) obtained from A092392 by taking every third column starting from column 0; x*c(x)^3 is the o.g.f. for A000245. - Peter Bala, Nov 24 2015

Examples

			Triangle begins
1,
2, 1,
6, 5, 1,
20, 21, 8, 1,
70, 84, 45, 11, 1,
252, 330, 220, 78, 14, 1,
924, 1287, 1001, 455, 120, 17, 1,
3432, 5005, 4368, 2380, 816, 171, 20, 1

Links

Crossrefs

Cf. A000245, A007318, A092392, A108080, A108081, A114422, A116382.

Programs

  • Magma
    /* As triangle */ [[Binomial(2*n+k, n+2*k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Nov 27 2015

Formula

Number triangle T(n,k) = Sum_{j = 0..n} binomial(n+k,j-k)*binomialC(n,j).
T(n,k) = binomial(2*n + k, n + 2*k). - Peter Bala, Nov 24 2015

A360144 a(n) = Sum_{k=0..n} binomial(2*n+3*k,n-k).

Original entry on oeis.org

1, 3, 14, 69, 344, 1721, 8621, 43206, 216570, 1085574, 5441294, 27272044, 136679882, 684959516, 3432431414, 17199626276, 86182614207, 431824008713, 2163629549132, 10840520569183, 54313805146415, 272122594209738, 1363372115057995, 6830627007245263
Offset: 0

Views

Author

Seiichi Manyama, Jan 27 2023

Keywords

Crossrefs

Cf. A001700, A032443, A108080, A360143.
Cf. A000108, A000344, A000984.

Programs

  • Maple
    A360144 := proc(n)
    add(binomial(2*n+3*k,n-k),k=0..n) ;
end proc:
seq(A360144(n),n=0..70) ; # R. J. Mathar, Mar 12 2023
  • PARI
    a(n) = sum(k=0, n, binomial(2*n+3*k, n-k));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/(sqrt(1-4*x)*(1-x*(2/(1+sqrt(1-4*x)))^5)))

Formula

G.f.: 1 / ( sqrt(1-4*x) * (1 - x * c(x)^5) ), where c(x) is the g.f. of A000108.
D-finite with recurrence +n*(697*n-7543)*a(n) +(697*n^2+23641*n-3800)*a(n-1) +2*(-32006*n^2+199879*n-255053)*a(n-2) +(283953*n^2-2288641*n+4072186)*a(n-3) +2*(-186566*n^2+1774989*n-4013515)*a(n-4) +(146221*n^2-1648033*n+4472550)*a(n-5) +(38223*n^2-307771*n+532906)*a(n-6) -10*(1511*n-6875)*(2*n-13)*a(n-7)=0. - R. J. Mathar, Mar 12 2023
a(n) = binomial(2*n, n)*hypergeom([1, (1+2*n)/3, 2*(1+n)/3, 1+2*n/3, -n], [(1+n)/4, (2+n)/4, (3+n)/4, 1+n/4], -3^3/4^4). - Stefano Spezia, Jun 17 2025

A360143 a(n) = Sum_{k=0..n} binomial(2*n+2*k,n-k).

Original entry on oeis.org

1, 3, 13, 59, 271, 1250, 5775, 26696, 123423, 570576, 2637306, 12187755, 56312089, 260134905, 1201493926, 5548533913, 25619837773, 118283258215, 546041467522, 2520515546083, 11633752319476, 53693477980816, 247798435809211, 1143547904185879, 5277058908767419
Offset: 0

Views

Author

Seiichi Manyama, Jan 27 2023

Keywords

Links

Crossrefs

Cf. A001700, A032443, A108080, A360144.
Cf. A000108, A000984, A002057.

Programs

  • Maple
    A360143 := proc(n)
    add(binomial(2*n+2*k,n-k),k=0..n) ;
end proc:
seq(A360143(n),n=0..70) ;# R. J. Mathar, Mar 12 2023
  • Mathematica
    Table[Sum[Binomial[2n+2k,n-k],{k,0,n}],{n,0,30}] (* Harvey P. Dale, Jul 23 2025 *)
  • PARI
    a(n) = sum(k=0, n, binomial(2*n+2*k, n-k));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/(sqrt(1-4*x)*(1-x*(2/(1+sqrt(1-4*x)))^4)))

Formula

G.f.: 1 / ( sqrt(1-4*x) * (1 - x * c(x)^4) ), where c(x) is the g.f. of A000108.
D-finite with recurrence +n*(n-7)*a(n) -(7*n-4)*(n-7)*a(n-1) +4*(n^2-13*n+17)*a(n-2) +(35*n^2-217*n+304)*a(n-3) -2*(n-2)*(7*n-29)*a(n-4) +4*(n-2)*(2*n-9)*a(n-5)=0. - R. J. Mathar, Mar 12 2023
a(n) = binomial(2*n, n)*hypergeom([1, -n, 1/2+n, 1+n], [(1+n)/3, (2+n)/3, 1+n/3], -4/27). - Stefano Spezia, Jun 17 2025
Showing 1-3 of 3 results.

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