The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A204579 Triangle read by rows: matrix inverse of the central factorial numbers T(2n, 2k) (A036969). 4
 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, 1017067024, -173721912, 14739153, -669188, 16422, -204, 1 (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 Table of n, a(n) for n=1..45. José L. Cereceda, Sums of powers of integers and the sequence A304330, arXiv:2405.05268 [math.GM], 2024. See p. 6. Mark W. Coffey, Matthew 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) T(n, k) = T(n-1, k-1) - (n-1)^2*T(n-1, k). (Recurrence equation.) 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) From Peter Luschny, Feb 29 2024: (Start) T(n, k) = [z^(2*k)] z^2*Product_{j=1..n-1} (z^2 - j^2). T(n, k) = (2*n)! * [t^k] [x^(2*n)] (w^sqrt(t) + w^(-sqrt(t)))/2 where w = (x/2 + sqrt(1 + (x/2)^2)^2. (End) EXAMPLE Triangle starts: [1] 1; [2] -1, 1; [3] 4, -5, 1; [4] -36, 49, -14, 1; [5] 576, -820, 273, -30, 1; [6] -14400, 21076, -7645, 1023, -55, 1; [7] 518400, -773136, 296296, -44473, 3003, -91, 1; [8] -25401600, 38402064, -15291640, 2475473, -191620, 7462, -140, 1; MAPLE # From Peter Luschny, Feb 29 2024: (Start) ogf := n -> local j; z^2*mul(z^2 - j^2, j = 1..n-1): Trow := n -> local k; seq(coeff(expand(ogf(n)), z, 2*k), k = 1..n): # Alternative: f := w -> (w^sqrt(t) + w^(-sqrt(t)))/2: egf := f((x/2 + sqrt(1 + (x/2)^2))^2): ser := series(egf, x, 20): cx := n -> coeff(ser, x, 2*n): Trow := n -> local k; seq((2*n)!*coeff(cx(n), t, k), k = 1..n): # (End) 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. A036969, A008955, A008275, A121408, A001044 (column 1), A101686 (alternating row sums), A234324 (central terms). 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified June 16 11:53 EDT 2024. Contains 373429 sequences. (Running on oeis4.)