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A111049 Triangle T(n,k), 0<=k<=n, read by rows, given by [1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0,0, ...] where DELTA is the operator defined in A084938. 1
1, 1, 1, 1, 3, 2, 1, 6, 9, 4, 1, 11, 27, 25, 8, 1, 20, 70, 100, 65, 16, 1, 37, 170, 330, 325, 161, 32, 1, 70, 399, 980, 1295, 966, 385, 64, 1, 135, 917, 2723, 4515, 4501, 2635, 2695, 897, 128, 1, 264, 2076, 7224, 14406, 17976, 14364, 7176, 2049 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

Table of n, a(n) for n=0..54.

FORMULA

T(n, k) = 2^(n-1)binomial(n-1, k-1)+binomial(n-1, k).

Sum_{k, 0<=k<=n} T(n, k) = 2^(n-1)*(1+2^(n-1)) = A063376(n-1) for n>=1.

From Peter Bala, Mar 20 2013: (Start)

O.g.f. : (1 - 2*t + x*t*(t-2) + x^2*t^2)/((1 - t*(1+x))*(1 - 2*t*(1+x))) = 1 + (1+x)*t + (1+3*x+2*x^2)*t^2 + ....

E.g.f. : (x + 2*exp((1+x)*t) + x*exp(2*t*(1+x)))/(2*(1+x)) = 1 + (1+x)*t + (1+3*x+2*x^2)*t^2/2! + ....

Recurrence equation: for n >= 1, T(n+1,k) = 2*T*n,k) + 2*T(n,k-1) - binomial(n,k). (End)

From Philippe Deléham, Oct 18 2013: (Start)

G.f. : (1 - 2*x - 2*x*y + x^2*y + x^2*y^2)/(1 - 3*x - 3*x*y + 2*x^2 + 4*x^2*y + 2*x^2*y^2).

T(n,k) = 3*T(n-1,k) + 3*T(n-1,k-1) - 2*T(n-2,k) - 4*T(n-2,k-1) - 2*T(n-2,k-2), T(0,0) = T(1,1) = T(1,0) = T(2,0) = 1, T(2,1) = 3, T(2,2) = 2, T(n,k) = 0 if k>n or if k<0. (End)

EXAMPLE

Rows begin:

1;

1, 1;

1, 3, 2;

1, 6, 9, 4;

1, 11, 27, 25, 8;

1, 20, 70, 100, 65, 16;

1, 37, 170, 330, 325, 161, 32;

1, 70, 399, 980, 1295, 966, 385, 64;

1, 135, 917, 2723, 4515, 4501, 2695, 897, 128;

1, 264, 2076, 7224, 14406, 17976, 14364, 7176, 2049, 256;

CROSSREFS

Cf. A002064, A006127.

Sequence in context: A227790 A181897 A212207 * A211955 A088617 A190909

Adjacent sequences:  A111046 A111047 A111048 * A111050 A111051 A111052

KEYWORD

easy,nonn,tabl

AUTHOR

Philippe Deléham, Oct 07 2005

STATUS

approved

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Last modified November 14 19:58 EST 2019. Contains 329128 sequences. (Running on oeis4.)