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.

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A096470 Triangle T(n,k), read by rows, formed by setting all entries in the zeroth column and in the main diagonal ((n,n) entries) to 1 and defining the rest of the entries by the recursion T(n,k) = T(n-1,k) - T(n,k-1).

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, -1, 2, 1, 1, -2, 4, -3, 1, 1, -3, 7, -10, 11, 1, 1, -4, 11, -21, 32, -31, 1, 1, -5, 16, -37, 69, -100, 101, 1, 1, -6, 22, -59, 128, -228, 329, -328, 1, 1, -7, 29, -88, 216, -444, 773, -1101, 1102, 1, 1, -8, 37, -125, 341, -785, 1558, -2659, 3761, -3760, 1, 1, -9, 46, -171, 512, -1297, 2855, -5514, 9275, -13035, 13036, 1
Offset: 0

Views

Author

Gerald McGarvey, Aug 12 2004

Keywords

Comments

If A(x,y) is the bivariate o.g.f. of a triangular array T(n,k) and B(x,y) is the bivariate o.g.f. of its mirror image T(n,n-k), then B(x,y) = A(x*y, y^(-1)) and A(x,y) = B(x*y, y^(-1)). - Petros Hadjicostas, Aug 08 2020

Examples

			From _Petros Hadjicostas_, Aug 08 2020: (Start)
Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  1,  1;
  1,  0,  1;
  1, -1,  2,   1;
  1, -2,  4,  -3,   1;
  1, -3,  7, -10,  11,    1;
  1, -4, 11, -21,  32,  -31,   1;
  1, -5, 16, -37,  69, -100, 101,    1;
  1, -6, 22, -59, 128, -228, 329, -328, 1;
  ... (End)

Crossrefs

Cf. A000108, A000124, A032357, A033297, A168377.

Programs

  • PARI
    T(n, k) = if ((k==0) || (n==k), 1, if ((n<0) || (k<0), 0, if (n>k, T(n-1, k) - T(n, k-1), 0)));
for(n=0, 10, for (k=0, n, print1(T(n, k), ", ")); print); \\ Petros Hadjicostas, Aug 08 2020

Formula

T(n,k) = T(n-1,k) - T(n,k-1) for 1 <= k <= n-1 with T(n,0) = 1 = T(n,n) for n >= 0.
The 2nd column is T(n,2) = A000124(n-2) for n >= 2 (Hogben's central polygonal numbers).
The "first subdiagonal" (unsigned) is |T(n,n-1)| = A032357(n-1) for n >= 1 (Convolution of Catalan numbers and powers of -1).
The "2nd subdiagonal" (unsigned) is |T(n,n-2)| = A033297(n) = Sum_{i=0..n-2} (-1)^i*C(n-1-i) for n >= 2, where C(n) are the Catalan numbers (A000108).
From Petros Hadjicostas, Aug 08 2020: (Start)
|T(n,k)| = |A168377(n,n-k)| for 0 <= k <= n.
Bivariate o.g.f.: (1 + y + x*y*c(-x*y))/((1 - x*y)*(1 - x + y)), where c(x) = 2/(1 + sqrt(1 - 4*x)) = o.g.f. of A000108.
Bivariate o.g.f. of |T(n,k)|: (1 - y - x*y*c(x*y))/((1 + x*y)*(1 - x - y)) + 2*x*y/(1 - x^2*y^2).
Bivariate o.g.f. of mirror image T(n,n-k): (1 + y + x*y*c(-x))/((1 - x)*(1 + y - x*y^2)).
Bivariate o.g.f. of |T(n,n-k)|: (1 - y + x*y*c(x))/((1 + x)*(1 - y + x*y^2)) + 2*x/(1 - x^2). (End)

Extensions

Offset changed to 0 by Petros Hadjicostas, Aug 08 2020

A271453 Triangle read by rows of coefficients of polynomials C_n(x) = Sum_{k=0..n} (2*k)!*(x - 1)^(n-k)/((k + 1)!*k!).

Original entry on oeis.org

1, 0, 1, 2, -1, 1, 3, 3, -2, 1, 11, 0, 5, -3, 1, 31, 11, -5, 8, -4, 1, 101, 20, 16, -13, 12, -5, 1, 328, 81, 4, 29, -25, 17, -6, 1, 1102, 247, 77, -25, 54, -42, 23, -7, 1, 3760, 855, 170, 102, -79, 96, -65, 30, -8, 1, 13036, 2905, 685, 68, 181, -175, 161, -95, 38, -9, 1, 45750, 10131, 2220, 617, -113, 356, -336, 256, -133, 47, -10, 1
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 09 2016

Keywords

Comments

The polynomials C_n(x) have generating function G(x,t) = (1 - sqrt(1 - 4*t))/(2*t*(1 + t - x*t)) = 1 + x*t + (x^2 - x + 2)*t^2 + (x^3 - 2*x^2 + 3*x + 3)*t^3 + ...
C_n(x) can be defined by the recurrence relation C_n(x) = (x - 1)*C_(n-1)(x) + (2n)!/((n + 1)!*n!), C_0(x) = 1 or the equivalent form C_n(x) = (x - 1)*C_(n-1)(x) + C_n(1), C_0(x) = 1.
C_n(x) can be defined as convolution of Catalan numbers and powers of (x - 1).
Discriminants of C_n(x) gives the sequence: 1, 1, -7, -543, 533489, 7080307052, -1318026434480736, -3526797951451513832247, 137992774365121594001729513153, ...
C_n(0) = A032357(n).
C_n(1) = C_n(x) - (x - 1)*C_(n-1)(x) = A000108(n).
C_n(2) = Sum_{m=0..n} C_1(m) = A014137(n).
C_n(3) = A014318(n).
C_n(5) = A000346(n).
C_n(6) = A046714(n).

Examples

			Triangle begins:
   1;
   0,  1;
   2, -1,  1;
   3,  3, -2,  1;
  11,  0,  5, -3,  1;
  31, 11, -5,  8, -4,  1;
  ...
The first few polynomials are:
  C_0(x) = 1;
  C_1(x) = x;
  C_2(x) = x^2 -   x   + 2;
  C_3(x) = x^3 - 2*x^2 + 3*x   + 3;
  C_4(x) = x^4 - 3*x^3 + 5*x^2         + 11;
  C_5(x) = x^5 - 4*x^4 + 8*x^3 - 5*x^2 + 11*x + 31;
  ...

Links

Crossrefs

Cf. A000108, A130595.

Programs

  • Mathematica
    CoefficientList[RecurrenceTable[{c[0] == 1, c[n] == (x - 1) c[n - 1] + CatalanNumber[n]}, c, {n, 11}], x]
T[n_, n_]:= 1; T[n_, 0]:= (-1)^n*Sum[CatalanNumber[k]*(-1)^k, {k, 0, n}]; T[n_, k_]:= T[n - 1, k - 1] - T[n - 1, k]; Table[T[n, k], {n, 0, 5}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 04 2018 *)
  • PARI
    {T(n, k) = if(k==n, 1, if(k==0, sum(j=0,n, (-1)^(n-j)*(2*j)!/(j!*(j+1)!)), T(n-1, k-1) - T(n-1, k))) };
for(n=0, 10, for(k=0, n, print1(T(n,k), ", "))) \\ G. C. Greubel, Nov 04 2018

Formula

For triangle: T(n,n)=1, T(n,0) = Sum_{k=0..n} (-1)^(n-k)*(2*k)!/(k! * (k+1)!), T(n, k) = T(n-1, k-1) - T(n-1, k). - G. C. Greubel, Nov 04 2018
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