A288871 Triangle t needed for the e.g.f.s of the column sequences of A288870 with leading zeros.

5, 9, 14, 15, 22, 36, 25, 34, 52, 88, 43, 54, 76, 120, 208, 77, 90, 116, 168, 272, 480, 143, 158, 188, 248, 368, 608, 1088, 273, 290, 324, 392, 528, 800, 1344, 2432, 531, 550, 588, 664, 816, 1120, 1728, 2944, 5376, 1045, 1066, 1108, 1192, 1360, 1696, 2368, 3712, 6400, 11776

Offset: 1

Author

Wolfdieter Lang, Jun 21 2017

Keywords

• nonn
• tabl
• easy

Comments

See the triangle T = A288870. The e.g.f. of the sequence of column k (k >= 0) without the leading k zeros is E(k, x) = (2*k+1)*exp(2*x) + exp(x). In order to get the e.g.f. for the column k sequence with the leading k zeros one has to integrate k times for k >=1; but this will first generate unwanted fractional numbers for the first k entries (when no integration constants are taken into account). These rational polynomials of degree k to be subtracted are S(k, x) = 2^(-k)* Sum_{m=1..k} t(k,m)*x^(m-1)/(m-1)! if k >=1.

Examples

			The triangle t begins:
k\m     1    2    3    4    5    6    7    8    9    10 ...
1:      5
2:      9   14
3:     15   22   36
4:     25   34   52   88
5:     43   54   76  120  208
6:     77   90  116  168  272  480
7:    143  158  188  248  368  608  108
8:    273  290  324  392  528  800 1344 2432
9:    531  550  588  664  816 1120 1728 2944 5376
10:  1045 1066 1108 1192 1360 1696 2368 3712 6400 11776
...
k = 1: E(1, x) = 3*exp(2*x) + exp(x) generates exponentially: 4, 7, 13, 25, 49, ..., the column k = 1 of T = A288870 without leading zero. Integration gives (without integration constant) (3/2)*exp(2*x) + exp(x), generating 5/2, 4, 7, 13, 25, 49, ..., therefore 5/2 = 2^(-1)* t(1,1)*x^(1-1)/(1-1)!= 2^(-1)*5*x^0 = 5/2.
Column o.g.f. for m=2: G(2, x) = 1/(1-2*x) + 4*x/(1-x)^2 + 2/(1-x) - (3 + 2^1*4*x) = 2*x^2*(7-17*x+8*x^2)/((1 - 2*x)*( 1 - x)^2).
		

Crossrefs

Cf. A288870.

Formula

t(k, m) = 2^k + k*2^m + 2^(m-1), k >= m >= 1, otherwise 0.

O.g.f. column m: G(m, x) =x*(2*x)^(m-1)*(3 - 5*x + 2*(1 - 3*x + 2*x^2)*m)/((1-x)^2*(1-2*x)).

O.g.f. G(m, x) = 1/(1-2*x) + 2^m*x/(1-x)^2 + 2^(m-1)/(1-x) - Subt(m ,x), with

Subt(m, x) = Sum_{k=0..m-1} A288870(m-1, k)*(2*x)^k.