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 1-5 of 5 results.

A107884 Matrix cube of triangle A107876; equals the product of triangular matrices: A107876^3 = A107862^-1*A107873.

Original entry on oeis.org

1, 3, 1, 6, 3, 1, 16, 9, 3, 1, 63, 37, 12, 3, 1, 351, 210, 67, 15, 3, 1, 2609, 1575, 498, 106, 18, 3, 1, 24636, 14943, 4701, 975, 154, 21, 3, 1, 284631, 173109, 54298, 11100, 1689, 211, 24, 3, 1, 3909926, 2381814, 745734, 151148, 22518, 2688, 277, 27, 3, 1
Offset: 0

Views

Author

Paul D. Hanna, Jun 04 2005

Keywords

Comments

Column 0 is A107885.
Column 1 is A107886.
Column 2 equals A107887.
Column 3 equals SHIFT_LEFT(A107878), where A107878 is column 2 of A107876.
Column 4 equals A107888.

Examples

			G.f. for column 0:
1 = T(0,0)*(1-x)^3 + T(1,0)*x*(1-x)^3 + T(2,0)*x^2*(1-x)^4 + T(3,0)*x^3*(1-x)^6 + T(4,0)*x^4*(1-x)^9 + T(5,0)*x^5*(1-x)^13 + ...
  = 1*(1-x)^3 + 3*x*(1-x)^3 + 6*x^2*(1-x)^4 + 16*x^3*(1-x)^6 + 63*x^4*(1-x)^9 + 351*x^5*(1-x)^13 + ...
G.f. for column 1:
1 = T(1,1)*(1-x)^3 + T(2,1)*x*(1-x)^4 + T(3,1)*x^2*(1-x)^6 + T(4,1)*x^3*(1-x)^9 + T(5,1)*x^4*(1-x)^13 + T(6,1)*x^5*(1-x)^18 + ...
  = 1*(1-x)^3 + 3*x*(1-x)^4 + 9*x^2*(1-x)^6 + 37*x^3*(1-x)^9 + 210*x^4*(1-x)^13 + 1575*x^5*(1-x)^18 + ...
Triangle begins:
       1;
       3,      1;
       6,      3,     1;
      16,      9,     3,     1;
      63,     37,    12,     3,    1;
     351,    210,    67,    15,    3,   1;
    2609,   1575,   498,   106,   18,   3,  1;
   24636,  14943,  4701,   975,  154,  21,  3, 1;
  284631, 173109, 54298, 11100, 1689, 211, 24, 3, 1;
  ...

Crossrefs

Cf. A107862, A107870, A107873, A107867, A107876, A107880, A107884, A107885, A107886, A107887, A107888.

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, 3];
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))^(3+(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)^(3 + (k+j)*(k+j-1)/2 - k*(k-1)/2).

A107887 Column 2 of triangle A107884.

Original entry on oeis.org

1, 3, 12, 67, 498, 4701, 54298, 745734, 11911221, 217418722, 4471886340, 102454974993, 2589782600870, 71643147090159, 2154145374733176, 69981625464827605, 2443741571641202568, 91309620200404008348
Offset: 0

Views

Author

Paul D. Hanna, Jun 04 2005

Keywords

Examples

			G.f. = 1 + 3*x + 12*x^2 + 67*x^3 + 498*x^4 + 4701*x^5 + 54298*x^6 + ...
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. A107884, A107885, A107886, A107888.

Programs

  • Mathematica
    a[ n_, k_: 2, j_: 1] := 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+1)*(k+2)/2)),n)}

Formula

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

A107885 Column 0 of triangle A107884.

Original entry on oeis.org

1, 3, 6, 16, 63, 351, 2609, 24636, 284631, 3909926, 62459868, 1140206571, 23453446130, 537365482536, 13583603874930, 375783149051267, 11299149697288503, 367080116280553305, 12818485840320445236
Offset: 0

Views

Author

Paul D. Hanna, Jun 04 2005

Keywords

Examples

			1 = 1*(1-x)^3 + 3*x*(1-x)^3 + 6*x^2*(1-x)^4 +
16*x^3*(1-x)^6 + 63*x^4*(1-x)^9 + 351*x^5*(1-x)^13 +...

Crossrefs

Cf. A107884, A107886, A107887, A107888.

Programs

  • PARI
    {a(n)=polcoeff(1-sum(k=0,n-1,a(k)*x^k*(1-x+x*O(x^n))^(3+k*(k-1)/2)),n)}

Formula

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

A107886 Column 1 of triangle A107884.

Original entry on oeis.org

1, 3, 9, 37, 210, 1575, 14943, 173109, 2381814, 38087355, 695745075, 14317460370, 328142173159, 8296618775100, 229557238129530, 6903176055689085, 224285333475911340, 7832574292981396104, 292678312428437482293
Offset: 0

Views

Author

Paul D. Hanna, Jun 04 2005

Keywords

Examples

			G.f. = 1 + 3*x + 9*x^2 + 37*x^3 + 210*x^4 + 1575*x^5 + 14943*x^6 + ...
1 = 1*(1-x)^3 + 3*x*(1-x)^4 + 9*x^2*(1-x)^6 + 37*x^3*(1-x)^9 + 210*x^4*(1-x)^13 + 1575*x^5*(1-x)^18 + ...

Crossrefs

Cf. A107884, A107885, A107887, A107888.

Programs

  • Mathematica
    a[ n_, k_: 3, 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))^(3+k*(k+1)/2)),n)

Formula

G.f.: 1 = Sum_{k>=0} a(k)*x^k*(1-x)^(3 + 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..3} i+1,
a(3) = Sum_{i=1..3}Sum_{j=1..i+1} j+2,
a(4) = Sum_{i=1..3}Sum_{j=1..i+1}Sum_{k=1..j+2} k+3,
a(5) = Sum_{i=1..3}Sum_{j=1..i+1}Sum_{k=1..j+2}Sum_{l=1..k+3} l+4. (End)

A107889 Triangular matrix T, read by rows, that satisfies: [T^-k](n,k) = -T(n,k-1) for n >= k > 0, or, equivalently, (column k of T^-k) = -SHIFT_LEFT(column k-1 of T) when zeros above the diagonal are ignored. Also, matrix inverse of triangle A107876.

Original entry on oeis.org

1, -1, 1, 0, -1, 1, 0, -1, -1, 1, 0, -3, -2, -1, 1, 0, -15, -9, -3, -1, 1, 0, -106, -61, -18, -4, -1, 1, 0, -975, -550, -154, -30, -5, -1, 1, 0, -11100, -6195, -1689, -310, -45, -6, -1, 1, 0, -151148, -83837, -22518, -4005, -545, -63, -7, -1, 1, 0, -2401365, -1326923, -353211, -61686, -8105, -875, -84, -8, -1, 1
Offset: 0

Views

Author

Paul D. Hanna, Jun 05 2005

Keywords

Comments

SHIFT_LEFT(column 1) = -A107878.
SHIFT_LEFT(column 2) = -A107883.
SHIFT_LEFT(column 3) = -A107888.

Examples

			G.f. for column 1:
1 = T(1,1)*(1-x)^-1 + T(2,1)*x*(1-x)^0 + T(3,1)*x^2*(1-x)^2 + T(4,1)*x^3*(1-x)^5 + T(5,1)*x^4*(1-x)^9 + T(6,1)*x^5*(1-x)^14 + ...
  = 1*(1-x)^-1 - 1*x*(1-x)^0 - 1*x^2*(1-x)^2 - 3*x^3*(1-x)^5 - 15*x^4*(1-x)^9 - 106*x^5*(1-x)^14 - 975*x^6*(1-x)^20 + ...
G.f. for column 2:
1 = T(2,2)*(1-x)^-1 + T(3,2)*x*(1-x)^1 + T(4,2)*x^2*(1-x)^4 + T(5,2)*x^3*(1-x)^8 + T(6,2)*x^4*(1-x)^13 + T(7,2)*x^5*(1-x)^19 + ...
  = 1*(1-x)^-1 - 1*x*(1-x)^1 - 2*x^2*(1-x)^4 - 9*x^3*(1-x)^8 - 61*x^4*(1-x)^13 - 550*x^5*(1-x)^19 - 6195*x^6*(1-x)^26 + ...
Triangle begins:
   1;
  -1,      1;
   0,     -1,     1;
   0,     -1,    -1,     1;
   0,     -3,    -2,    -1,    1;
   0,    -15,    -9,    -3,   -1,   1;
   0,   -106,   -61,   -18,   -4,  -1,  1;
   0,   -975,  -550,  -154,  -30,  -5, -1,  1;
   0, -11100, -6195, -1689, -310, -45, -6, -1, 1;
  ...

Crossrefs

Cf. A107876, A107880, A107884.

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 = Inverse[Inverse[A107862].A107867];
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))^(-1+(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)^(-1 + (k+j)*(k+j-1)/2 - k*(k-1)/2).
Showing 1-5 of 5 results.

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