A136555 Square array, read by antidiagonals, where T(n,k) = binomial(2^k + n-1, k).

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

Offset: 0

Author

Paul D. Hanna, Jan 07 2008

Keywords

• nonn
• tabl

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.

Examples

			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.
		

Links

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

Crossrefs

Rows: A014070, A136505, A136506, A136556.

Diagonals: A060690, A132683, A132684.

Cf. A136557 (antidiagonal sums).

Cf. A132625.

Programs

Magma

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

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 *)

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

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)