A110552 A triangular array related to A077028 and distributing the values of A007582.

1, 1, 2, 1, 5, 4, 1, 10, 17, 8, 1, 19, 51, 49, 16, 1, 36, 134, 196, 129, 32, 1, 69, 330, 650, 645, 321, 64, 1, 134, 783, 1940, 2575, 1926, 769, 128, 1, 263, 1813, 5411, 8995, 8981, 5383, 1793, 256, 1, 520, 4124, 14392, 28742, 35896, 28700, 14344, 4097, 512, 1, 1033, 9252, 36948, 86142, 129150, 129108, 86052, 36873, 9217, 1024

Offset: 1

Author

Alford Arnold, Jul 26 2005

Keywords

• nonn
• tabl

Comments

Let T(r,c) be the array A077028. Fill 2^k numbers in Gaussian templates conforming to the row lengths determined by T(r,c). A110552 results from summing the numbers on each row.

Examples

			The filled templates begin
1
.1
.2
..1
..2.3
..4
....1
....2.3.5
....4.6.7
....8
therefore the sequence begins
1
1 2
1 5 4
1 10 17 8
...
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
• P. Barry, A Note on a Family of Generalized Pascal Matrices Defined by Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.5.4.

Crossrefs

Cf. A077028, A007051, A007582. A111049.

Programs

Mathematica

T[n_, k_] := Binomial[n - 2, k - 1] + 2^(n - 1)*Binomial[n - 2, k - 2]; Table[T[n, k], {n, 1, 20}, {k, 1, n}] // Flatten (* G. C. Greubel, Aug 31 2017 *)

PARI

for(n=1,20, for(k=1,n, print1(binomial(n - 2, k - 1) + 2^(n - 1)*binomial(n - 2, k - 2), ", "))) \\ G. C. Greubel, Aug 31 2017

Formula

Table entries appear to be given by T(n,k) = binomial(n-2,k-1) + 2^(n-1)*binomial(n-2,k-2), n,k >= 1, leading to the e.g.f. (exp((1+x)*u) - 1)*(x*exp((1+x)*u) + x + 2)/(2*(1+x)^2) = u + (1+2*x)*u^2/2! + (1+5*x+4*x^2)*u^3/3! + .... Cf. A111049. - Peter Bala, Jul 27 2012