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

Author

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

Keywords

• nonn
• tabl

Comments

The first term of the m-th row is 2^m-1.

Examples

			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,
...
		

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.

Crossrefs

Cf. A174531.

Programs

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 *)

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

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).