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).
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
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Jan 07 2008
STATUS
approved