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
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.
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