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A155761 Riordan array (c(2*x^2), x*c(2*x^2)) where c(x) is the g.f. of A000108. 3
1, 0, 1, 2, 0, 1, 0, 4, 0, 1, 8, 0, 6, 0, 1, 0, 20, 0, 8, 0, 1, 40, 0, 36, 0, 10, 0, 1, 0, 112, 0, 56, 0, 12, 0, 1, 224, 0, 224, 0, 80, 0, 14, 0, 1, 0, 672, 0, 384, 0, 108, 0, 16, 0, 1, 1344, 0, 1440, 0, 600, 0, 140, 0, 18, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Inverse of Riordan array (1/(1+2*x^2), x/(1+2*x^2)).

LINKS

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

FORMULA

T(n,k) = (1+(-1)^(n-k)) * ((k+1)/(n+1)) * binomial(n+1, (n-k)/2) * 2^((n-k-2)/2).

Sum_{k=0..n} T(n, k) = A126087(n).

T(n,k) = 2^((n-k)/2) * A053121(n,k). - Philippe Deléham, Feb 11 2009

Sum_{k=0..n} T(2*n-k, k) = A064062(n+1). - G. C. Greubel, Jun 06 2021

EXAMPLE

Triangle begins:

    1;

    0,   1;

    2,   0,   1;

    0,   4,   0,  1;

    8,   0,   6,  0,  1;

    0,  20,   0,  8,  0,  1;

   40,   0,  36,  0, 10,  0,  1;

    0, 112,   0, 56,  0, 12,  0, 1;

  224,   0, 224,  0, 80,  0, 14, 0, 1;

Production matrix begins as:

  0, 1;

  2, 0, 1;

  0, 2, 0, 1;

  0, 0, 2, 0, 1;

  0, 0, 0, 2, 0, 1;

  0, 0, 0, 0, 2, 0, 1;

  0, 0, 0, 0, 0, 2, 0, 1;

  0, 0, 0, 0, 0, 0, 2, 0, 1;

  0, 0, 0, 0, 0, 0, 0, 2, 0, 1;

  0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1;

MATHEMATICA

T[n_, k_]:= (1+(-1)^(n-k))*2^((n-k-2)/2)*((k+1)/(n+1))*Binomial[n+1, (n-k)/2];

Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 06 2021 *)

PROG

(Sage)

def A155761(n, k): return (1+(-1)^(n-k))*2^((n-k-2)/2)*((k+1)/(n+1))*binomial(n+1, (n-k)/2)

flatten([[A155761(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 06 2021

CROSSREFS

Cf. A064062, A126087 (row sums).

Sequence in context: A111959 A110109 A145973 * A067631 A134317 A123641

Adjacent sequences:  A155758 A155759 A155760 * A155762 A155763 A155764

KEYWORD

easy,nonn,tabl

AUTHOR

Paul Barry, Jan 26 2009

STATUS

approved

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Last modified October 27 12:59 EDT 2021. Contains 348276 sequences. (Running on oeis4.)