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 A204579 Triangle read by rows: matrix inverse of A036969. 3
 1, -1, 1, 4, -5, 1, -36, 49, -14, 1, 576, -820, 273, -30, 1, -14400, 21076, -7645, 1023, -55, 1, 518400, -773136, 296296, -44473, 3003, -91, 1, -25401600, 38402064, -15291640, 2475473, -191620, 7462, -140, 1, 1625702400, -2483133696 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS This is a signed version of A008955 with rows in reverse order. - Peter Luschny, Feb 04 2012 LINKS M. W. Coffey, M. C. Lettington, On Fibonacci Polynomial Expressions for Sums of mth Powers, their implications for Faulhaber's Formula and some Theorems of Fermat, arXiv:1510.05402 [math.NT], 2015. FORMULA T(n,k) = (-1)^(n-k)*A008955(n, n-k). - Peter Luschny, Feb 05 2012 T(n,k) = Sum_{i=k-n..n-k} (-1)^(n-k+i)*s(n,k+i)*s(n,k-i) = Sum_{i=0..2*k} (-1)^(n+i)*s(n,i)*s(n,2*k-i), where s(n,k) are Stirling numbers of the first kind, A048994. - Mircea Merca, Apr 07 2012 From Peter Bala, Aug 29 2012: (Start) Recurrence equation: T(n,k) = T(n-1,k-1) - (n-1)^2*T(n-1,k). Let E(x) = cosh(sqrt(2*x)) = Sum_{n >= 0} x^n/{(2*n)!/2^n} and L(x) = 2*{arcsinh(sqrt(x/2))}^2 = Sum_{n >=1} (-1)^n*(n-1)!^2*x^n/{(2*n)!/2^n}. L(x) is the compositional inverse of E(x) - 1. A generating function for the triangle is E(t*L(x)) = 1 + t*x + t*(-1 + t)*x^2/6 + t*(4 - 5*t + t^2)*x^3/90 + ..., where the sequence of denominators [1,1,6,90,...] is given by (2*n)!/2^n. Cf. A008275 with generating function exp(t*log(1+x)). The e.g.f. is E(t*L(x^2/2)) = cosh(2*sqrt(t)*arcsinh(x/2)) = 1 + t*x^2/2! + t*(t-1)*x^4/4! + t*(t-1)*(t-4)*x^6/6! + .... (End) EXAMPLE Padding A036969 with zeros yields the infinite square matrix [ 1  0  0  0 ...] [ 1  1  0  0 ...] [ 1  5  1  0 ...] [ 1 21 14  1 ...] with inverse [  1  0  0  0 ...] [ -1  1  0  0 ...] [  4 -5  1  0 ...] [-36 49 -14 1 ...]. MATHEMATICA rows = 10; t[n_, k_] := 2*Sum[j^(2*n)*(-1)^(k - j)/((k - j)!*(k + j)!), {j, 1, k}]; T = Table[t[n, k], {n, 1, rows}, {k, 1, rows}] // Inverse; Table[T[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 14 2018 *) PROG (PARI) select(concat(Vec(matrix(10, 10, n, k, T(n, k)/*from A036969*/)~^-1)), x->x) (Sage) def A204579(n, k) : return (-1)^(n-k)*A008955(n, n-k)     for n in (0..7) : print [A204579(n, k) for k in (0..n)] # Peter Luschny, Feb 05 2012 CROSSREFS Cf. A008955. A008275, A121408. Sequence in context: A110519 A286796 A286718 * A113095 A157784 A274615 Adjacent sequences:  A204576 A204577 A204578 * A204580 A204581 A204582 KEYWORD sign,tabl AUTHOR M. F. Hasler, Feb 03 2012 EXTENSIONS Typo in data corrected by Peter Luschny, Feb 05 2012 STATUS approved

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Last modified February 18 09:42 EST 2019. Contains 320249 sequences. (Running on oeis4.)