A185139 Triangle T(n,k) = Sum_{i=1..n} 2^(i-1)*C(n+2*k-i-1, k-1), 1 <= k <= n.

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

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: A261836 A301937 A373866 * A300786 A182241 A281178

Adjacent sequences: A185136 A185137 A185138 * A185140 A185141 A185142

Keyword

nonn,tabl

Author

Vladimir Shevelev and Peter J. C. Moses, Feb 04 2012

Status

approved