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.

Previous Showing 21-23 of 23 results.

A121435 Matrix inverse of triangle A122175, where A122175(n,k) = C( k*(k+1)/2 + n-k, n-k) for n>=k>=0.

Original entry on oeis.org

1, -1, 1, 1, -2, 1, -2, 5, -4, 1, 7, -19, 18, -7, 1, -37, 104, -106, 49, -11, 1, 268, -766, 809, -406, 110, -16, 1, -2496, 7197, -7746, 4060, -1210, 216, -22, 1, 28612, -82910, 90199, -48461, 15235, -3032, 385, -29, 1, -391189, 1136923, -1244891, 678874, -220352, 46732, -6699, 638, -37, 1
Offset: 0

Views

Author

Paul D. Hanna, Aug 27 2006

Keywords

Examples

			Triangle begins:
1;
-1, 1;
1, -2, 1;
-2, 5, -4, 1;
7, -19, 18, -7, 1;
-37, 104, -106, 49, -11, 1;
268, -766, 809, -406, 110, -16, 1;
-2496, 7197, -7746, 4060, -1210, 216, -22, 1;
28612, -82910, 90199, -48461, 15235, -3032, 385, -29, 1;
-391189, 1136923, -1244891, 678874, -220352, 46732, -6699, 638, -37, 1; ...

Crossrefs

Cf. A098568, A107876; unsigned columns: A107877, A107882.

Programs

  • PARI
    /* Matrix Inverse of A122175 */ T(n,k)=local(M=matrix(n+1,n+1,r,c,if(r>=c,binomial((c-1)*(c-2)/2+r-1,r-c)))); return((M^-1)[n+1,k+1])
  • 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+1)), n-k)

Formula

(1) T(n,k) = A121434(n-1,k) - A121434(n-1,k+1).
(2) T(n,k) = (-1)^(n-k)*[A107876^(k*(k+1)/2 + 1)](n,k); i.e., column k equals signed column k of matrix power A107876^(k*(k+1)/2 + 1).
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 + 1);
(4) 1 = Sum_{j>=0} T(j+k,k)*x^j*(1+x)^( j*(j-1)/2) + j*k + k*(k+1)/2 + 1).

A121437 Matrix inverse of triangle A122177, where A122177(n,k) = C( k*(k+1)/2 + n-k + 2, n-k) for n>=k>=0.

Original entry on oeis.org

1, -3, 1, 6, -4, 1, -16, 14, -6, 1, 63, -62, 33, -9, 1, -351, 365, -215, 72, -13, 1, 2609, -2790, 1731, -642, 143, -18, 1, -24636, 26749, -17076, 6696, -1664, 261, -24, 1, 284631, -311769, 202356, -81963, 21684, -3831, 444, -31, 1, -3909926, 4305579, -2822991, 1166310, -320515, 60768, -8012, 713, -39, 1
Offset: 0

Views

Author

Paul D. Hanna, Aug 27 2006

Keywords

Examples

			Triangle begins:
  1;
  -3, 1;
  6, -4, 1;
  -16, 14, -6, 1;
  63, -62, 33, -9, 1;
  -351, 365, -215, 72, -13, 1;
  2609, -2790, 1731, -642, 143, -18, 1;
  -24636, 26749, -17076, 6696, -1664, 261, -24, 1;
  284631, -311769, 202356, -81963, 21684, -3831, 444, -31, 1; ...

Crossrefs

Cf. A098568, A107876, A122177, A107885.

Programs

  • PARI
    /* Matrix Inverse of A122177 */ T(n,k)=local(M=matrix(n+1,n+1,r,c,if(r>=c,binomial((c-1)*(c-2)/2+r+1,r-c)))); return((M^-1)[n+1,k+1])
  • 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+3)), n-k)

Formula

(1) T(n,k) = A121436(n-1,k) - A121436(n-1,k+1).
(2) T(n,k) = (-1)^(n-k)*[A107876^(k*(k+1)/2 + 3)](n,k); i.e., column k equals signed column k of A107876^(k*(k+1)/2 + 3).
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 + 3);
(4) 1 = Sum_{j>=0} T(j+k,k)*x^j*(1+x)^( j*(j-1)/2) + j*k + k*(k+1)/2 + 3).
From Benedict W. J. Irwin, Nov 26 2016: (Start)
Conjecture: The sequence (column 2 of triangle) 14, -62, 365, -2790, 26749, ... is described by a series of nested sums:
14 = Sum_{i=1..4} (i+1),
-62 = -Sum_{i=1..4} (Sum_{j=1..i+1} (j+2)),
365 = Sum_{i=1..4} (Sum_{j=1..i+1} (Sum_{k=1..j+2} (k+3))),
-2790 = -Sum_{i=1..4} (Sum_{j=1..i+1} (Sum_{k=1..j+2} (Sum_{l=1..k+3} (l+4)))). (End)

A300954 Number of Dyck paths whose sequence of ascent lengths is exactly n+1, n+2, ..., 2n.

Original entry on oeis.org

1, 1, 3, 26, 425, 10647, 365512, 16067454, 864721566, 55202528425, 4083666929771, 343854336973368, 32493430569907125, 3406873823160467912, 392619681705581846700, 49342834390595374213214, 6717520607597479710109299, 984991858956314599670220717, 154785386247352261724279606367
Offset: 0

Views

Author

Alois P. Heinz, Mar 16 2018

Keywords

Comments

Dyck paths counted by a(n) have semilength (3*n^2 + n)/2 = A005449(n) and length A049451(n).

Examples

			a(0) = 1: the empty path.
a(1) = 1: uudd.
a(2) = 3: uuuduuuudddddd, uuudduuuuddddd, uuuddduuuudddd.

Links

Crossrefs

Main diagonal of A107876.
Cf. A005449, A049451.

Programs

  • Maple
    a:= proc(m) option remember; local b; b:=
      proc(n, i) option remember; `if`(i>=2*m, 1,
        add(b(n+i-j, i+1), j=1..n+i))
      end; b(0, m+1)
    end:
seq(a(n), n=0..20);
  • Mathematica
    a[m_] := a[m] = Module[{b}, b[n_, i_] := b[n, i] = If[i >= 2m, 1, Sum[b[n + i - j, i + 1], {j, 1, n + i}]]; b[0, m + 1]];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jun 02 2018, from Maple *)

Formula

a(n) = A107876(2n,n).
Previous Showing 21-23 of 23 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.