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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

Table of n, a(n) for n=1..38.

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-Fran├žois 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.)