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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 December 1 23:26 EST 2023. Contains 367503 sequences. (Running on oeis4.)