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 1-4 of 4 results.

A107880 Matrix square of triangle A107876; equals matrix product of triangles: A107876^2 = A107862^-1*A107870 = A107867^-1*A107873.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 7, 5, 2, 1, 26, 19, 7, 2, 1, 141, 104, 37, 9, 2, 1, 1034, 766, 268, 61, 11, 2, 1, 9693, 7197, 2496, 550, 91, 13, 2, 1, 111522, 82910, 28612, 6195, 982, 127, 15, 2, 1, 1528112, 1136923, 391189, 83837, 12977, 1596, 169, 17, 2, 1, 24372513, 18141867, 6230646, 1326923, 202494, 24206, 2424, 217, 19, 2, 1
Offset: 0

Views

Author

Paul D. Hanna, Jun 04 2005

Keywords

Comments

Column 0 is A107881. Column 1 is A107882. Column 3 equals A107883. Column 2 equals SHIFT_LEFT(A107877), where A107877 is column 1 of A107876.

Examples

			G.f. for column 0:
1 = T(0,0)*(1-x)^2 + T(1,0)*x*(1-x)^2 + T(2,0)*x^2*(1-x)^3 + T(3,0)*x^3*(1-x)^5 + T(4,0)*x^4*(1-x)^8 + T(5,0)*x^5*(1-x)^12 +...
= 1*(1-x)^2 + 2*x*(1-x)^2 + 3*x^2*(1-x)^3 + 7*x^3*(1-x)^5 + 26*x^4*(1-x)^8 + 141*x^5*(1-x)^12 +...
G.f. for column 1:
1 = T(1,1)*(1-x)^2 + T(2,1)*x*(1-x)^3 + T(3,1)*x^2*(1-x)^5 + T(4,1)*x^3*(1-x)^8 + T(5,1)*x^4*(1-x)^12 + T(6,1)*x^5*(1-x)^17 +...
= 1*(1-x)^2 + 2*x*(1-x)^3 + 5*x^2*(1-x)^5 + 19*x^3*(1-x)^8 + 104*x^4*(1-x)^12 + 766*x^5*(1-x)^17 +...
Triangle T begins:
  1;
  2,1;
  3,2,1;
  7,5,2,1;
  26,19,7,2,1;
  141,104,37,9,2,1;
  1034,766,268,61,11,2,1;
  9693,7197,2496,550,91,13,2,1;
  111522,82910,28612,6195,982,127,15,2,1;
  ...

Crossrefs

Cf. A107862, A107870, A107873, A107867, A107876, A107884, A107887.

Programs

  • Mathematica
    max = 10;
A107862 = Table[Binomial[If[n < k, 0, n*(n-1)/2-k*(k-1)/2 + n - k], n - k], {n, 0, max}, {k, 0, max}];
A107867 = Table[Binomial[If[n < k, 0, n*(n-1)/2-k*(k-1)/2 + n-k+1], n - k], {n, 0, max}, {k, 0, max}];
T = MatrixPower[Inverse[A107862].A107867, 2];
Table[T[[n+1, k+1]], {n, 0, max}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 31 2024 *)
  • PARI
    {T(n,k)=polcoeff(1-sum(j=0,n-k-1, T(j+k,k)*x^j*(1-x+x*O(x^n))^(2+(k+j)*(k+j-1)/2-k*(k-1)/2)),n-k)}

Formula

G.f. for column k: 1 = Sum_{j>=0} T(k+j, k)*x^j*(1-x)^(2+(k+j)*(k+j-1)/2-k*(k-1)/2).

A107882 Column 1 of triangle A107880.

Original entry on oeis.org

1, 2, 5, 19, 104, 766, 7197, 82910, 1136923, 18141867, 330940109, 6803936050, 155839142185, 3938383850350, 108934529005948, 3275059508166297, 106388204134734785, 3714826559490125850, 138796913898027894261
Offset: 0

Views

Author

Paul D. Hanna, Jun 04 2005

Keywords

Examples

			G.f. = 1 + 2*x + 5*x^2 + 19*x^3 + 104*x^4 + 766*x^5 + 7197*x^6 + 82910*x^7 + ...
1 = 1*(1-x)^2 + 2*x*(1-x)^3 + 5*x^2*(1-x)^5 +
19*x^3*(1-x)^8 + 104*x^4*(1-x)^12 + 766*x^5*(1-x)^17 +...

Crossrefs

Cf. A107880, A107881, A107883.

Programs

  • Mathematica
    a[ n_, k_: 2, j_: 0] := If[ n < 1, Boole[n >= 0], a[ n, k, j] = Sum[ a[ n - 1, i, j + 1], {i, k + j}]]; (* Michael Somos, Nov 26 2016 *)
  • PARI
    {a(n)=polcoeff(1-sum(k=0,n-1,a(k)*x^k*(1-x+x*O(x^n))^(2+k*(k+1)/2)),n)}

Formula

G.f.: 1 = Sum_{k>=0} a(k)*x^k*(1-x)^(2 + k*(k+1)/2).
From Benedict W. J. Irwin, Nov 26 2016: (Start)
Conjecture: a(n) can be expressed with a series of nested sums,
a(2) = Sum_{i=1..2} i+1,
a(3) = Sum_{i=1..2}Sum_{j=1..i+1} j+2,
a(4) = Sum_{i=1..2}Sum_{j=1..i+1}Sum_{k=1..j+2} k+3,
a(5) = Sum_{i=1..2}Sum_{j=1..i+1}Sum_{k=1..j+2}Sum_{l=1..k+3} l+4. (End)

A107883 Column 3 of triangle A107880.

Original entry on oeis.org

1, 2, 9, 61, 550, 6195, 83837, 1326923, 24078588, 493309850, 11271757335, 284379843234, 7856320956198, 235986714918110, 7660827258318780, 267365373971139600, 9985779421324740445, 397508459931685273305
Offset: 0

Views

Author

Paul D. Hanna, Jun 04 2005

Keywords

Examples

			1 = 1*(1-x)^2 + 2*x*(1-x)^5 + 9*x^2*(1-x)^9 +
61*x^3*(1-x)^14 + 550*x^4*(1-x)^20 + 6195*x^5*(1-x)^27 +...

Crossrefs

Cf. A107880, A107881, A107882.

Programs

  • Mathematica
    a[ n_, k_: 0, j_: 2] := If[n < 1, Boole[n >= 0], a[ n, k, j] = Sum[ a[ n - 1, i, j + 1], {i, k + j}]]; (* Michael Somos, Nov 26 2016 *)
  • PARI
    {a(n)=polcoeff(1-sum(k=0,n-1,a(k)*x^k*(1-x+x*O(x^n))^((k+2)*(k+3)/2-1)),n)}

Formula

G.f.: 1 = Sum_{k>=0} a(k)*x^k*(1-x)^((k+2)*(k+3)/2 - 1).

A121436 Matrix inverse of triangle A122176, where A122176(n,k) = C( k*(k+1)/2 + n-k + 1, n-k) for n>=k>=0.

Original entry on oeis.org

1, -2, 1, 3, -3, 1, -7, 9, -5, 1, 26, -37, 25, -8, 1, -141, 210, -155, 60, -12, 1, 1034, -1575, 1215, -516, 126, -17, 1, -9693, 14943, -11806, 5270, -1426, 238, -23, 1, 111522, -173109, 138660, -63696, 18267, -3417, 414, -30, 1, -1528112, 2381814, -1923765, 899226, -267084, 53431, -7337, 675, -38, 1
Offset: 0

Views

Author

Paul D. Hanna, Aug 27 2006

Keywords

Examples

			Triangle begins:
1;
-2, 1;
3, -3, 1;
-7, 9, -5, 1;
26, -37, 25, -8, 1;
-141, 210, -155, 60, -12, 1;
1034, -1575, 1215, -516, 126, -17, 1;
-9693, 14943, -11806, 5270, -1426, 238, -23, 1;
111522, -173109, 138660, -63696, 18267, -3417, 414, -30, 1;
-1528112, 2381814, -1923765, 899226, -267084, 53431, -7337, 675, -38, 1; ...

Links

Crossrefs

Cf. A098568, A107876; unsigned columns: A107881, A107886.

Programs

  • PARI
    /* Matrix Inverse of A122176 */
{T(n,k)=local(M=matrix(n+1,n+1,r,c,if(r>=c,binomial((c-1)*(c-2)/2+r,r-c)))); return((M^-1)[n+1,k+1])}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))
  • PARI
    /* Obtain by G.F. */
{T(n,k)=polcoeff(1-sum(j=0, n-k-1, T(j+k,k)*x^j/(1-x+x*O(x^n))^(j*(j+1)/2+j*k+k*(k+1)/2+2)), n-k)}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

Formula

(1) T(n,k) = A121435(n-1,k) - A121435(n-1,k+1).
(2) T(n,k) = (-1)^(n-k)*[A107876^(k*(k+1)/2 + 2)](n,k);
i.e., column k equals signed column k of A107876^(k*(k+1)/2 + 2).
G.f.s for column k:
(3) 1 = Sum_{j>=0} T(j+k,k)*x^j/(1-x)^( j*(j+1)/2) + j*k + k*(k+1)/2 + 2);
(4) 1 = Sum_{j>=0} T(j+k,k)*x^j*(1+x)^( j*(j-1)/2) + j*k + k*(k+1)/2 + 2).
Showing 1-4 of 4 results.

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