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 A126671 Triangle read by rows: row n (n>=0) has g.f. Sum_{i=1..n} n!*x^i*(1+x)^(n-i)/(n+1-i). 12
 0, 0, 1, 0, 1, 3, 0, 2, 7, 11, 0, 6, 26, 46, 50, 0, 24, 126, 274, 326, 274, 0, 120, 744, 1956, 2844, 2556, 1764, 0, 720, 5160, 16008, 28092, 30708, 22212, 13068, 0, 5040, 41040, 147120, 304464, 401136, 351504, 212976, 109584, 0, 40320 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 COMMENTS The first nonzero column gives the factorial numbers, which are Stirling_1(*,1), the rightmost diagonal gives Stirling_1(*,2), so this triangle may be regarded as interpolating between the first two columns of the Stirling numbers of the first kind. This is a slice (the right-hand wall) through the infinite square pyramid described in the link. The other three walls give A007318 and A008276 (twice). The coefficients of the A165674 triangle are generated by the asymptotic expansion of the higher order exponential integral E(x,m=2,n). The a(n) formulas for the coefficients in the right hand columns of this triangle lead to Wiggen's triangle A028421 and their o.g.f.s. lead to the sequence given above. Some right hand columns of the A165674 triangle are A080663, A165676, A165677, A165678 and A165679. - Johannes W. Meijer, Oct 07 2009 LINKS N. J. A. Sloane, Notes on Carlo Wood's Polynomials FORMULA Recurrence: T(n,0) = 0; for n>=0, i>=1, T(n+1,i) = (n+1)*T(n,i) + n!*binomial(n,i). E.g.f.: x*log(1-(1+x)*y)/(x*y-1)/(1+x). - Vladeta Jovovic, Feb 13 2007 EXAMPLE Triangle begins: 0, 0, 1, 0, 1, 3, 0, 2, 7, 11, 0, 6, 26, 46, 50, 0, 24, 126, 274, 326, 274, 0, 120, 744, 1956, 2844, 2556, 1764, 0, 720, 5160, 16008, 28092, 30708, 22212, 13068, 0, 5040, 41040, 147120, 304464, 401136, 351504, 212976, 109584, 0, 40320, 367920, 1498320, 3582000, 5562576, 5868144, 4292496, 2239344, 1026576, ... MAPLE for n from 1 to 15 do t1:=add( n!*x^i*(1+x)^(n-i)/(n+1-i), i=1..n); series(t1, x, 100); lprint(seriestolist(%)); od: MATHEMATICA Join[{{0}}, Reap[For[n = 1, n <= 15, n++, t1 = Sum[n!*x^i*(1+x)^(n-i)/(n+1-i), {i, 1, n}]; se = Series[t1, {x, 0, 100}]; Sow[CoefficientList[se, x]]]][[2, 1]]] // Flatten (* Jean-François Alcover, Jan 07 2014, after Maple *) CROSSREFS Columns give A000142, A108217, A126672; diagonals give A000254, A067318, A126673. Row sums give A126674. Alternating row sums give A000142. See A126682 for the full pyramid of coefficients of the underlying polynomials. Cf. A165674, A028421, A080663, A165676, A165677, A165678 and A165679. - Johannes W. Meijer, Oct 07 2009 Sequence in context: A302728 A302528 A303410 * A209437 A325447 A240660 Adjacent sequences:  A126668 A126669 A126670 * A126672 A126673 A126674 KEYWORD nonn,tabl AUTHOR N. J. A. Sloane and Carlo Wood (carlo(AT)alinoe.com), Feb 13 2007 STATUS approved

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Last modified April 19 13:00 EDT 2021. Contains 343114 sequences. (Running on oeis4.)