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 A289192 A(n,k) = n! * Laguerre(n,-k); square array A(n,k), n>=0, k>=0, read by antidiagonals. 13
 1, 1, 1, 1, 2, 2, 1, 3, 7, 6, 1, 4, 14, 34, 24, 1, 5, 23, 86, 209, 120, 1, 6, 34, 168, 648, 1546, 720, 1, 7, 47, 286, 1473, 5752, 13327, 5040, 1, 8, 62, 446, 2840, 14988, 58576, 130922, 40320, 1, 9, 79, 654, 4929, 32344, 173007, 671568, 1441729, 362880 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 LINKS Alois P. Heinz, Antidiagonals n = 0..140, flattened Eric Weisstein's World of Mathematics, Laguerre Polynomial Wikipedia, Laguerre polynomials FORMULA A(n,k) = n! * Sum_{i=0..n} k^i/i! * binomial(n,i). E.g.f. of column k: exp(k*x/(1-x))/(1-x). A(n, k) = (-1)^n*KummerU(-n, 1, -k). - Peter Luschny, Feb 12 2020 EXAMPLE Square array A(n,k) begins: :   1,    1,    1,     1,     1,     1, ... :   1,    2,    3,     4,     5,     6, ... :   2,    7,   14,    23,    34,    47, ... :   6,   34,   86,   168,   286,   446, ... :  24,  209,  648,  1473,  2840,  4929, ... : 120, 1546, 5752, 14988, 32344, 61870, ... MAPLE A:= (n, k)-> n! * add(binomial(n, i)*k^i/i!, i=0..n): seq(seq(A(n, d-n), n=0..d), d=0..12); MATHEMATICA A[n_, k_] := n! * LaguerreL[n, -k]; Table[A[n - k, k], {n, 0, 9}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, May 05 2019 *) (* As a triangle read by ascending antidiagonals: *) a[n_, k_] := (-1)^n HypergeometricU[-n, 1, -k]; Table[Round[a[n - k, k]], {n, 0, 9}, {k, 0, n}] // Flatten (* Peter Luschny, Feb 12 2020 *) PROG (Python) from sympy import binomial, factorial as f def A(n, k): return f(n)*sum([binomial(n, i)*k**i/f(i) for i in range(n + 1)]) for n in range(13): print [A(k, n - k) for k in range(n + 1)] # Indranil Ghosh, Jun 28 2017 CROSSREFS Columns k=0-10 give: A000142, A002720, A087912, A277382, A289147, A289211, A289212, A289213, A289214, A289215, A289216. Rows n=0-2 give: A000012, A000027(k+1), A008865(k+2). Main diagonal gives A277373. Sequence in context: A299500 A330141 A007441 * A111933 A144304 A122941 Adjacent sequences:  A289189 A289190 A289191 * A289193 A289194 A289195 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Jun 28 2017 STATUS approved

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Last modified May 30 05:08 EDT 2020. Contains 334712 sequences. (Running on oeis4.)