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A215241 Unsigned matrix inverse of triangle A214398, as a triangle read by rows n>=1. 5
1, 1, 1, 3, 4, 1, 18, 26, 9, 1, 172, 256, 99, 16, 1, 2313, 3489, 1416, 264, 25, 1, 40626, 61696, 25650, 5120, 575, 36, 1, 887326, 1352518, 569772, 117980, 14450, 1098, 49, 1, 23282964, 35566368, 15099042, 3193728, 410850, 34608, 1911, 64, 1, 715540140, 1094499820 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,4

LINKS

Paul D. Hanna, Table of n, a(n) for n = 1..1081

FORMULA

G.f.: x*y/(1-x*y) = Sum_{n>=1} Sum_{k=1..n} T(n,k)*x^n*y^k/(1+x)^(n^2).

G.f. of column k: 1 = Sum_{n>=k} T(n,k)*x^(n-k)/(1+x)^(n^2).

Column 1 forms A177447.

Row sums form A133316.

EXAMPLE

Triangle begins:

1;

1, 1;

3, 4, 1;

18, 26, 9, 1;

172, 256, 99, 16, 1;

2313, 3489, 1416, 264, 25, 1;

40626, 61696, 25650, 5120, 575, 36, 1;

887326, 1352518, 569772, 117980, 14450, 1098, 49, 1;

23282964, 35566368, 15099042, 3193728, 410850, 34608, 1911, 64, 1;

715540140, 1094499820, 466865280, 100049120, 13259705, 1186857, 73696, 3104, 81, 1;

25259729071, 38670814348, 16545757617, 3575578368, 482959200, 44908920, 2994684, 143616, 4779, 100, 1; ...

The matrix inverse is a signed version of triangle A214398:

1;

-1, 1;

1, -4, 1;

-1, 10, -9, 1;

1, -20, 45, -16, 1;

-1, 35, -165, 136, -25, 1;

1, -56, 495, -816, 325, -36, 1;

-1, 84, -1287, 3876, -2925, 666, -49, 1; ...

in which the g.f. of column k is 1/(1+x)^(k^2) for k>=1.

ILLUSTRATE G.F. OF COLUMNS:

k=1: 1 = 1/(1+x) + 1*x/(1+x)^4 + 3*x^2/(1+x)^9 + 18*x^3/(1+x)^16 + 172*x^4/(1+x)^25 + 2313*x^5/(1+x)^36 + 40626*x^6/(1+x)^49 +...

k=2: 1 = 1/(1+x)^4 + 4*x/(1+x)^9 + 26*x^2/(1+x)^16 + 256*x^3/(1+x)^25 + 3489*x^4/(1+x)^36 + 61696*x^5/(1+x)^49 +...

k=3: 1 = 1/(1+x)^9 + 9*x/(1+x)^16 + 99*x^2/(1+x)^25 + 1416*x^3/(1+x)^36 + 25650*x^4/(1+x)^49 +...

k=4: 1 = 1/(1+x)^16 + 16*x/(1+x)^25 + 264*x^2/(1+x)^36 + 5120*x^3/(1+x)^49 +...

PROG

(PARI) {T(n, k)=local(M=matrix(n, n, r, c, binomial(c^2+r-c-1, r-c))); (-1)^(n-k)*(M^-1)[n, k]}

for(n=1, 12, for(k=1, n, print1(T(n, k), ", ")); print(""))

CROSSREFS

Cf. A177447 (column 1), A215242 (column 2), A215243 (column 3); A133316 (row sums).

Cf. A214398 (unsigned matrix inverse).

Sequence in context: A136228 A356145 A154829 * A055133 A342186 A113084

Adjacent sequences:  A215238 A215239 A215240 * A215242 A215243 A215244

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Aug 06 2012

STATUS

approved

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Last modified October 3 22:17 EDT 2022. Contains 357237 sequences. (Running on oeis4.)