A173111 -id:A173111 - cp's OEIS frontend

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

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

A173108 Triangle, A000110 in every column > 0, shifted down twice.

Original entry on oeis.org

1, 1, 2, 1, 5, 1, 15, 2, 1, 52, 5, 1, 203, 15, 2, 1, 877, 52, 5, 1, 4140, 203, 15, 2, 1, 21147, 877, 52, 5, 1, 115975, 4140, 203, 15, 2, 1, 678570, 21147, 877, 52, 5, 1, 4213597, 115975, 4140, 203, 15, 2, 1, 27644437, 678570, 21147, 877, 52, 5, 1

Offset: 0

Gary W. Adamson, Feb 09 2010

Row sums = A173109: (1, 1, 3, 6, 18, 58, 221, 935, ...).

Let the triangle = M. Then lim_{n->oo} M^n = A173110: (1, 1, 3, 6, 20, 60, ...).

			First few rows of the triangle:
       1;
       1;
       2,    1;
       5,    1;
      15,    2,   1;
      52,    5,   1;
     203,   15,   2,  1;
     877,   52,   5,  1;
    4140,  203,  15,  2, 1;
   21147,  877,  52,  5, 1;
  115975, 4140, 203, 15, 2, 1;
  ...

Cf. A000110, A173109, A173110, A173111.

T[n_, k_] := BellB[n - 2 k];
Table[T[n, k], {n, 0, 10}, {k, 0, Quotient[n, 2]}] // Flatten (* Jean-François Alcover, Apr 22 2022 *)

B(n) = sum(k=0, n, stirling(n, k, 2)); \\ A000110
tabf(nn) = for (n=0, nn, for(k=0, n\2, print1(B(n-2*k), ", "));); \\ Michel Marcus, Nov 19 2022

Bell sequence in every column, for columns > 0, shifted down twice.

Keyword tabf and more terms from Michel Marcus, Nov 19 2022

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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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A173108 Triangle, A000110 in every column > 0, shifted down twice.

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