A059594 Convolution triangle based on A008619 (positive integers repeated).

1, 1, 1, 2, 2, 1, 2, 5, 3, 1, 3, 8, 9, 4, 1, 3, 14, 19, 14, 5, 1, 4, 20, 39, 36, 20, 6, 1, 4, 30, 69, 85, 60, 27, 7, 1, 5, 40, 119, 176, 160, 92, 35, 8, 1, 5, 55, 189, 344, 376, 273, 133, 44, 9, 1, 6, 70, 294, 624, 820, 714, 434

Offset

0,4

Comments

In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Bell-subgroup of the Riordan-group.

The G.f. for the row polynomials p(n,x) = Sum_{m=0..n} a(n,m)*x^m is 1/((1-z^2)*(1-z)-x*z).

The column sequences are A008619(n); A006918(n); A038163(n-2), n >= 2; A038164(n-3), n >= 3; A038165(n-4), n >= 4; A038166(n-5), n >= 5; A059595(n-6), n >= 6; A059596(n-7), n >= 7; A059597(n-8), n >= 8; A059598(n-9), n >= 9; A059625(n-10), n >= 10 for m=0..10.

The sequence of row sums is A006054(n+2).

From Gary W. Adamson, Aug 14 2016: (Start)

The sequence can be generated by extracting the descending antidiagonals of an array formed by taking powers of the natural integers with repeats, (1, 1, 2, 2, 3, 3, ...), as follows:

1, 1, 2, 2, 3, 3, ...

1, 2, 5, 8, 14, 20, ...

1, 3, 9, 19, 39, 69, ...

1, 4, 14, 36, 85, 176, ...

...

Row sums of the triangle = (1, 2, 5, 11, 25, 56, ...), the INVERT transform of (1, 1, 2, 2, 3, 3, ...). (End)

Links

Table of n, a(n) for n=0..61.

Formula

a(n, m) := a(n-1, m) + (-(n-m+1)*a(n, m-1) + 3*(n+2*m)*a(n-1, m-1))/(8*m), n >= m >= 1; a(n, 0) := floor((n+2)/2) = A008619(n), n >= 0; a(n, m) := 0 if n < m.

G.f.for column m >= 0: ((x/((1-x^2)*(1-x)))^m)/((1-x^2)*(1-x)).

T(n,m) = Sum_{k=0..n-m} (Sum_{j=0..k} binomial(j, n-m-3*k+2*j)*(-1)^(j-k)*binomial(k,j))*binomial(m+k,m). - Vladimir Kruchinin, Dec 14 2011

Recurrence: T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) - T(n-3,k) with T(0,0) = 1. - Philippe Deléham, Feb 23 2012

Example

{1}; {1,1}; {2,2,1}; {2,5,3,1}; ...
Fourth row polynomial (n=3): p(3,x)= 2 + 5*x + 3*x^2 + x^3.

Mathematica

t[n_, m_] := Sum[Sum[Binomial[j, n-m-3*k+2*j]*(-1)^(j-k)*Binomial[k, j], {j, 0, k}]*Binomial[m+k, m], {k, 0, n-m}]; Table[t[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, May 27 2013, after Vladimir Kruchinin *)

Prog

(Maxima)
T(n, m):=sum((sum(binomial(j, n-m-3*k+2*j)*(-1)^(j-k)*binomial(k, j), j, 0, k)) *binomial(m+k, m), k, 0, n-m); /* Vladimir Kruchinin, Dec 14 2011 */

Crossrefs

Sequence in context: A325182 A215959 A209555 * A183760 A125678 A091562

Adjacent sequences: A059591 A059592 A059593 * A059595 A059596 A059597

Keyword

nonn,easy,tabl

Author

Wolfdieter Lang, Feb 02 2001

Status

approved