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 A185139 Triangle T(n,k) = Sum_{i=1..n} 2^(i-1)*C(n+2*k-i-1, k-1), 1 <= k <= n. 1
 1, 3, 10, 7, 25, 91, 15, 56, 210, 792, 31, 119, 456, 1749, 6721, 63, 246, 957, 3718, 14443, 56134, 127, 501, 1969, 7722, 30251, 118456, 463828, 255, 1012, 4004, 15808, 62322, 245480, 966416, 3803648, 511, 2035, 8086, 32071, 127024, 502588, 1987096, 7852453, 31020445, 1023, 4082, 16263 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS The first term of the m-th row is 2^m-1. LINKS G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened V. Shevelev and P. Moses, On a sequence of polynomials with hypothetically integer coefficients arXiv:1112.5715 [math.NT], 2011. FORMULA 2*T_n(k) = T_(n-1)(k+1) + C(n+2*k-1,k). T_n(k) = T_(n-2)(k+1) + C(n+2*k-1,k). T_n(k) = 2*T_(n-1)(k) + C(n+2*k-2,k-1). T_n(k+1) = 4*T_n(k) - (n/k)*C(n+2*k-1,k-1). EXAMPLE Triangle begins 1, 3,     10, 7,     25,    91, 15,    56,    210,  792, 31,    119,   456,  1749,  6721, 63,    246,   957,  3718,  14443,  56134, 127,   501,   1969, 7722,  30251,  118456, 463828, 255,   1012,  4004, 15808, 62322,  245480, 966416,  3803648, 511,   2035,  8086, 32071, 127024, 502588, 1987096, 7852453, 31020445, ... MATHEMATICA Table[Sum[2^(j - 1)*Binomial[n + 2*k - j - 1, k - 1], {j, 1, n}], {n,    1, 50}, {k, 1, n}] // Flatten (* G. C. Greubel, Jun 23 2017 *) PROG (PARI) for(n=1, 20, for(k=1, n, print1(sum(j=1, n, 2^(j-1)*binomial(n+2*k-j-1, k-1)), ", "))) \\ G. C. Greubel, Jun 23 2017 CROSSREFS Cf. A174531. Sequence in context: A195922 A261836 A301937 * A300786 A182241 A033152 Adjacent sequences:  A185136 A185137 A185138 * A185140 A185141 A185142 KEYWORD nonn,tabl AUTHOR Vladimir Shevelev and Peter J. C. Moses, Feb 04 2012 STATUS approved

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Last modified September 16 16:17 EDT 2021. Contains 347473 sequences. (Running on oeis4.)