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 1-2 of 2 results.

A131321 Triangle read by rows: A168561^2.

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 0, 4, 0, 1, 5, 0, 6, 0, 1, 0, 14, 0, 8, 0, 1, 13, 0, 27, 0, 10, 0, 1, 0, 46, 0, 44, 0, 12, 0, 1, 34, 0, 107, 0, 65, 0, 14, 0, 1, 0, 145, 0, 204, 0, 90, 0, 16, 0, 1, 89, 0, 393, 0, 345, 0, 119, 0, 18, 0, 1, 0, 444, 0, 854, 0, 538, 0, 152, 0, 20, 0, 1
Offset: 0

Views

Author

Gary W. Adamson, Jun 28 2007

Keywords

Comments

Left border, nonzero terms = odd indexed Fibonacci numbers: (1, 2, 5, 13, ...). Next column, nonzero terms = A030267: (1, 4, 14, 46, 145, ...). Row sums = A131322: (1, 1, 3, 5, 12, 23, 51, ...).
Riordan array (f(x),x*f(x)) where f(x) = (1-x^2)/(1-3*x^2+x^4). Aerated version of triangle in A188137. - Philippe Deléham, Jan 26 2012

Examples

			First few rows of the triangle are:
   1;
   0,  1;
   2,  0,  1;
   0,  4,  0,  1;
   5,  0,  6,  0,  1;
   0, 14,  0,  8,  0,  1;
  13,  0, 27,  0, 10,  0,  1;
  ...

Crossrefs

Cf. A049310, A030267, A168561, A188137.

Programs

  • Maple
    F:= (n, k)-> coeff(combinat[fibonacci](n+1, x), x, k):
T:= (n, k)-> add(F(n, j)*F(j, k), j=0..n):
seq(seq(T(n, k), k=0..n), n=0..14);  # Alois P. Heinz, Dec 12 2019

Formula

A168561 squared, as an infinite lower triangular matrix.

A188365 a(n) = n! * [x^n] exp((1 - 2*x)/(1 - 3*x + x^2) - 1).

Original entry on oeis.org

1, 1, 5, 43, 505, 7421, 130501, 2668975, 62197073, 1626103225, 47116726021, 1498191224531, 51855200633545, 1940384578283893, 78042911672096645, 3357060094366363351, 153771739817047383841, 7471843888639307665265, 383835896530177022152453, 20783664252941721959512315
Offset: 0

Views

Author

Vladimir Kruchinin, Mar 28 2011

Keywords

Links

Crossrefs

Cf. A001519, A188137.

Programs

  • Maple
    gf := exp((1 - 2*x)/(1 - 3*x + x^2) - 1): ser := series(gf, x, 22):
seq(k!*coeff(ser, x, k), k=0..19); # Peter Luschny, Jul 30 2020
  • PARI
    f(n,m) = sum(k=m, n, binomial(n-1,k-1) * sum(i=ceil((k-m)/2), k-m, binomial(i,k-m-i)*binomial(m+i-1,m-1))); \\ A188137
a(n) = if (n, n!*sum(k=1, n, f(n,k)/k!), 1); \\ Michel Marcus, Jul 30 2020
  • PARI
    my(x='x+O('x^25)); Vec(serlaplace(exp((1-2*x)/(1-3*x+x^2)-1))) \\ Joerg Arndt, Jul 30 2020

Formula

E.g.f.: exp((1 - 2*x)/(1 - 3*x + x^2) - 1) = exp(G(x) - 1) where G(x) is the o.g.f. of A001519.
a(n) = n! * Sum_{k=1..n} A188137(n,k)/k!, n>0, a(0)=1.

Extensions

More terms from Michel Marcus, Jul 30 2020
Showing 1-2 of 2 results.

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