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

1,3

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.

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.

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

Example

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
...

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 *)

Prog

(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

Crossrefs

Cf. A077028, A007051, A007582. A111049.

Sequence in context: A238241 A299444 A371300 * A129161 A103415 A054456

Adjacent sequences: A110549 A110550 A110551 * A110553 A110554 A110555

Keyword

nonn,tabl

Author

Alford Arnold, Jul 26 2005

Status

approved