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 A350470 Array read by ascending antidiagonals. T(n, k) = J(k, n) where J are the Jacobsthal polynomials. 5
 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 5, 1, 1, 1, 7, 9, 11, 1, 1, 1, 9, 13, 29, 21, 1, 1, 1, 11, 17, 55, 65, 43, 1, 1, 1, 13, 21, 89, 133, 181, 85, 1, 1, 1, 15, 25, 131, 225, 463, 441, 171, 1, 1, 1, 17, 29, 181, 341, 937, 1261, 1165, 341, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,9 LINKS Table of n, a(n) for n=0..65. FORMULA T(n, k) = Sum_{j=0..k} binomial(k - j, j)*(2*n)^j. T(n, k) = ((1+s)^(k+1) - (1-s)^(k+1)) / (2^(k+1)*s) where s = sqrt(8*n + 1). T(n, k) = [x^k] (1 / (1 - x - 2*n*x^2)). T(n, k) = hypergeom([1/2 - k/2, -k/2], [-k], -8*n). EXAMPLE Array starts: n\k 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ... --------------------------------------------------------------------- [0] 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... A000012 [1] 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, ... A001045 [2] 1, 1, 5, 9, 29, 65, 181, 441, 1165, 2929, ... A006131 [3] 1, 1, 7, 13, 55, 133, 463, 1261, 4039, 11605, ... A015441 [4] 1, 1, 9, 17, 89, 225, 937, 2737, 10233, 32129, ... A015443 [5] 1, 1, 11, 21, 131, 341, 1651, 5061, 21571, 72181, ... A015446 [6] 1, 1, 13, 25, 181, 481, 2653, 8425, 40261, 141361, ... A053404 [7] 1, 1, 15, 29, 239, 645, 3991, 13021, 68895, 251189, ... A350468 [8] 1, 1, 17, 33, 305, 833, 5713, 19041, 110449, 415105, ... A168579 [9] 1, 1, 19, 37, 379, 1045, 7867, 26677, 168283, 648469, ... A350469 A005408 | A082108 | A016813 A014641 MAPLE J := (n, x) -> add(2^k*binomial(n - k, k)*x^k, k = 0..n): seq(seq(J(k, n-k), k = 0..n), n = 0..10); MATHEMATICA T[n_, k_] := Hypergeometric2F1[(1 - k)/2, -k/2, -k, -8 n]; Table[T[n, k], {n, 0, 9}, {k, 0, 9}] // TableForm (* or *) T[n_, k_] := With[{s = Sqrt[8*n+1]}, ((1+s)^(k+1) - (1-s)^(k+1)) / (2^(k+1)*s)]; Table[Simplify[T[n, k]], {n, 0, 9}, {k, 0, 9}] // TableForm PROG (PARI) T(n, k) = ([1, 2; k, 0]^n)[1, 1] ; export(T) for(k = 0, 9, print(parvector(10, n, T(n - 1, k)))) CROSSREFS Rows: A000012, A001045, A006131, A015441, A015443, A015446, A053404, A350468, A168579, A350469. Columns: A000012, A005408, A016813, A082108, A014641. Cf. A350467 (main diagonal), A352361 (Fibonacci polynomials), A352362 (Lucas polynomials). Sequence in context: A254101 A349025 A348963 * A277604 A112475 A347232 Adjacent sequences: A350467 A350468 A350469 * A350471 A350472 A350473 KEYWORD nonn,tabl AUTHOR Peter Luschny, Mar 19 2022 STATUS approved

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Last modified March 4 02:53 EST 2024. Contains 370522 sequences. (Running on oeis4.)