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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
José L. Cereceda, Sums of powers of integers and the sequence A304330, arXiv:2405.05268 [math.GM], 2024. See p. 6.
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
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 June 16 11:53 EDT 2024. Contains 373429 sequences. (Running on oeis4.)