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 A288871 Triangle t needed for the e.g.f.s of the column sequences of A288870 with leading zeros. 1
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 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. LINKS 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. EXAMPLE 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. Sequence in context: A218981 A030772 A314799 * A325694 A314800 A314801 Adjacent sequences:  A288868 A288869 A288870 * A288872 A288873 A288874 KEYWORD nonn,tabl,easy AUTHOR Wolfdieter Lang, Jun 21 2017 STATUS approved

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Last modified August 1 10:37 EDT 2021. Contains 346385 sequences. (Running on oeis4.)