cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A107102 Matrix inverse of A100862.

Original entry on oeis.org

1, -1, 1, 2, -3, 1, -7, 12, -6, 1, 37, -67, 39, -10, 1, -266, 495, -310, 95, -15, 1, 2431, -4596, 3000, -1010, 195, -21, 1, -27007, 51583, -34566, 12320, -2660, 357, -28, 1, 353522, -680037, 463981, -171766, 39795, -6062, 602, -36, 1, -5329837, 10306152, -7124454, 2709525, -658791, 108927, -12432, 954
Offset: 0

Views

Author

Paul D. Hanna, May 21 2005

Keywords

Comments

Column 0 is signed A001515 (Bessel polynomial). Column 1 is A107103. Row sums are zeros for n>0. Absolute row sums form A107104, which equals 2*A043301(n-1) for n>0.
The row polynomials p_n(x) of this entry are (-1)^n B_n(1-x), where B_n(x) are the modified Carlitz-Bessel polynomials of A001497, e,g, (-1)^2 B_2(1-x) = (1-x) + (1-x)^2 = 2 - 3 x + x^2 = p_2(x). - Tom Copeland, Oct 10 2016

Examples

			Triangle begins:
1;
-1,1;
2,-3,1;
-7,12,-6,1;
37,-67,39,-10,1;
-266,495,-310,95,-15,1;
2431,-4596,3000,-1010,195,-21,1;
-27007,51583,-34566,12320,-2660,357,-28,1; ...
and is the matrix inverse of A100862:
1;
1,1;
1,3,1;
1,6,6,1;
1,10,21,10,1;
1,15,55,55,15,1; ...

Crossrefs

Cf. A100862, A001515, A043301, A107103, A107104.
Cf. A001497.

Programs

  • PARI
    {T(n,k)=local(X=x+x*O(x^n),Y=y+y*O(y^n));(matrix(n+1,n+1,m,j,if(m>=j, (m-1)!*polcoeff(polcoeff(exp(X+Y*X^2/2+X*Y),m-1,x),j-1,y)))^-1)[n+1,k+1]}

Formula

E.g.f.: exp((1-y)*(1-sqrt(1+2*x))). [Vladeta Jovovic, Dec 13 2008]

A107104 Absolute row sums of triangle A107102, which is the matrix inverse of A100862.

Original entry on oeis.org

1, 2, 6, 26, 154, 1182, 11254, 128522, 1715802, 26251118, 453132214, 8714516538, 184817376154, 4285657717694, 107880712446966, 2929921866938858, 85399256991014746, 2659096654189212558, 88091786616208073398
Offset: 0

Views

Author

Paul D. Hanna, May 21 2005

Keywords

Links

Crossrefs

Cf. A100862, A043301, A107102, A107103.

Programs

  • PARI
    a(n)=if(n==0,1,2^n*sum(k=0,n-1,(n+k-1)!/(n-k-1)!/k!/4^k))
  • PARI
    a(n)=local(X=x+x*O(x^n),Y=y+y*O(y^n)); sum(k=0,n,abs((matrix(n+1,n+1,m,j,if(m>=j, (m-1)!*polcoeff(polcoeff(exp(X+Y*X^2/2+X*Y),m-1,x),j-1,y)))^-1)[n+1,k+1]))

Formula

a(n) = 2*A043301(n-1) = 2^n*Sum_{k=0..n-1}(n+k-1)!/(n-k-1)!/k!/4^k for n>0, with a(0) = 1.
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.