A173108 -id:A173108 - cp's OEIS frontend

A173109 Row sums of triangle A173108.

1, 1, 3, 6, 18, 58, 221, 935, 4361, 22082, 120336, 700652, 4333933, 28345089, 195233255, 1411303634, 10675375402, 84276173438, 692752181561, 5917018378495, 52416910416933, 480786834535246, 4559132648864256, 44632792689619592, 450518001943669545

Offset: 0

Gary W. Adamson, Feb 09 2010

nonn

			a(5) = 58 = 52 + 5 + 1.

Alois P. Heinz, Table of n, a(n) for n = 0..576

Cf. A000110, A087650, A173108.

b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-j)*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> add(b(n-2*k), k=0..iquo(n, 2)):
seq(a(n), n=0..25);  # Alois P. Heinz, Jun 17 2021

a[n_] := Sum[BellB[n-2k], {k, 0, Quotient[n, 2]}];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Apr 22 2022 *)

G.f.: G(0)/(1-x^2)/(1+x) where G(k) =  1 - 2*x*(k+1)/((2*k+1)*(2*x*k-1) - x*(2*k+1)*(2*k+3)*(2*x*k-1)/(x*(2*k+3) - 2*(k+1)*(2*x*k+x-1)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 04 2013.
G.f.: ( G(0) - 1 )/(1-x^2) where G(k) =  1 + (1-x)/(1-x*k)/(1-x/(x+(1-x)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 22 2013
a(n+1) - a(n) = A087650(n+1). - Vladimir Reshetnikov, Oct 29 2015
a(n) = Sum_{k=0..floor(n/2)} A000110(n-2*k). - Alois P. Heinz, Jun 17 2021

More terms from Sergei N. Gladkovskii, Jan 04 2013

A173110 Given triangle A173108 = M, then A173110 = Lim_{n->inf.} M^n; the left-shifted vector considered as a sequence.

Original entry on oeis.org

1, 1, 3, 6, 20, 60, 230, 950, 4420, 22230, 120914, 702820, 4343860, 28393280, 195492054

Offset: 0

Gary W. Adamson, Feb 09 2010

nonn

Triangle A173108 as an infinite lower triangular matrix * A173110 = A173110;
i.e. the sequence remains unchanged.
Contribution from Gary W. Adamson, Jul 08 2010: (Start)
Let B(x) = (1 + x + 2x^2 + 5x^3 + 15x^4 + ...), Bell numbers, A000110; and
A(x) = (1 + x + 3x^2 + 6x^3 + 20x^4 + ...). Then A(x) = B(x) * B(x^2) *
B(x^4) * B(x^8) * ...; and B(x) = A(x)/A(x^2). (End)

A173108, A173111

Cf. A000110 [From Gary W. Adamson, Jul 08 2010]

Given triangle A173108 = M, then A173110 = Lim_{n->inf.} M^n; the left-shifted vector considered as a sequence

A173111 Triangle read by rows, A173108 * the diagonalized variant of A173110.

Original entry on oeis.org

1, 1, 2, 1, 5, 1, 15, 2, 3, 52, 5, 3, 203, 15, 6, 6, 877, 52, 15, 6, 4140, 203, 45, 12, 20, 21147, 877, 156, 30, 20, 115975, 4140, 609, 90, 40, 60, 678570, 21147, 2631, 312, 100, 60

Offset: 0

Gary W. Adamson, Feb 09 2010

Row sums = A173110: (1, 1, 3, 6, 20, 60, 230, 950, 4420, 22230,...).

			First few rows of the triangle =
1;
1;
2, 1;
5, 1;
15, 2, 3;
52, 5, 3;
203, 15, 6, 6;
877, 52, 15, 6;
4140, 203, 45, 12, 20;
21147, 877, 156, 30, 20;
115975, 4140, 609, 90, 40, 60;
678570, 21147, 2631, 312, 100, 60;
...
Example: row 7 = termwise products of (877, 52, 5, 1) and (1, 1, 3, 6) =
(877, 52, 15, 6); where (877, 52, 5, 1) = row 7 of triangle A173108, and
(1, 1, 3, 6) = the first four terms of sequence A173109.

A173108, A173109, A173110

Let triangle A173108 = Q, and M = an infinite lower triangular matrix with A173110 as the rightmost diagonal and the rest zeros. Triangle A173111 = Q*M.

cp's OEIS Frontend

A173109 Row sums of triangle A173108.

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A173110 Given triangle A173108 = M, then A173110 = Lim_{n->inf.} M^n; the left-shifted vector considered as a sequence.

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A173111 Triangle read by rows, A173108 * the diagonalized variant of A173110.

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