A215241 Unsigned matrix inverse of triangle A214398, as a triangle read by rows n >= 1.

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

Offset: 1

Author

Paul D. Hanna, Aug 06 2012

Keywords

• nonn
• tabl

Examples

			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;
  ...
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 + ...
		

Links

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

Crossrefs

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

Cf. A214398 (unsigned matrix inverse).

Programs

Mathematica

T[n_, k_] := Module[{M}, M = Table[Binomial[c^2 + r - c - 1, r - c], {r, 1, n}, {c, 1, n}]; (-1)^(n - k) Inverse[M][[n, k]]];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 05 2023, after PARI program *)

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(""))

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.