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A136555 Square array, read by antidiagonals, where T(n,k) = binomial(2^k + n-1, k). 16
1, 1, 1, 1, 2, 3, 1, 3, 6, 35, 1, 4, 10, 56, 1365, 1, 5, 15, 84, 1820, 169911, 1, 6, 21, 120, 2380, 201376, 67945521, 1, 7, 28, 165, 3060, 237336, 74974368, 89356415775, 1, 8, 36, 220, 3876, 278256, 82598880, 94525795200, 396861704798625, 1, 9, 45, 286, 4845, 324632, 90858768, 99949406400, 409663695276000, 6098989894499557055 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Let vector R_{n} equal row n of this array; then R_{n+1} = P * R_{n} for n>=0, where triangle P = A132625 such that row n+1 of P = row n of P^(2^n) with appended '1' for n>=0.

LINKS

G. C. Greubel, Antidiagonal rows n = 0..50, flattened

FORMULA

G.f. for row n: Sum_{i>=0} (1 + 2^i*x)^(n-1) * log(1 + 2^i*x)^i / i!.

From G. C. Greubel, Mar 14 2021: (Start)

For the square array:

    T(n, n) = A060690(n).

  T(n+1, n) = A132683(n),   T(n+2, n) = A132684(n).

T(2*n+1, n) = A132685(n),   T(2*n, n) = A132686(n).

T(3*n+2, n) = A132689(n), T(3*n+1, n) = A132688(n), T(3*n, n) = A132687(n).

For the number triangle:

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

Sum_{k=0..n} t(n,k) = Sum_{k=0..n} T(n-k, k) = A136557(n). (End)

EXAMPLE

Square array begins:

  1, 1,  3,  35, 1365, 169911,  67945521,  89356415775, ... A136556;

  1, 2,  6,  56, 1820, 201376,  74974368,  94525795200, ... A014070;

  1, 3, 10,  84, 2380, 237336,  82598880,  99949406400, ... A136505;

  1, 4, 15, 120, 3060, 278256,  90858768, 105637584000, ... A136506;

  1, 5, 21, 165, 3876, 324632,  99795696, 111600996000, ... ;

  1, 6, 28, 220, 4845, 376992, 109453344, 117850651776, ... ;

  1, 7, 36, 286, 5985, 435897, 119877472, 124397910208, ... ;

  1, 8, 45, 364, 7315, 501942, 131115985, 131254487936, ... ;

  ...

Form column vector R_{n} out of row n of this array;

then row n+1 can be generated from row n by:

R_{n+1} = P * R_{n} for n>=0,

where triangular matrix P = A132625 begins:

        1;

        1,      1;

        2,      1,     1;

       14,      4,     1,    1;

      336,     60,     8,    1,  1;

    25836,   2960,   248,   16,  1, 1;

  6251504, 454072, 24800, 1008, 32, 1, 1; ...

where row n+1 of P = row n of P^(2^n) with appended '1' for n>=0.

MAPLE

A136555:= (n, k) -> binomial(2^k +n-k-1, k); seq(seq(A136555(n, k), k=0..n), n=0..12); # G. C. Greubel, Mar 14 2021

MATHEMATICA

Table[Binomial[2^k +n-k-1, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 14 2021 *)

PROG

(PARI) T(n, k)=binomial(2^k+n-1, k)

(PARI) /* Coefficient of x^k in g.f. of row n: */ T(n, k)=polcoeff(sum(i=0, k, (1+2^i*x+x*O(x^k))^(n-1)*log((1+2^i*x)+x*O(x^k))^i/i!), k)

(Sage) flatten([[binomial(2^k +n-k-1, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 14 2021

(Magma) [Binomial(2^k +n-k-1, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 14 2021

CROSSREFS

Rows: A014070, A136505, A136506, A136556.

Diagonals: A060690, A132683, A132684.

Cf. A136557 (antidiagonal sums).

Cf. A132625.

Sequence in context: A271702 A292915 A271700 * A343627 A188107 A174014

Adjacent sequences:  A136552 A136553 A136554 * A136556 A136557 A136558

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Jan 07 2008

STATUS

approved

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Last modified June 24 17:34 EDT 2021. Contains 345418 sequences. (Running on oeis4.)