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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A210391 Number A(n,k) of semistandard Young tableaux over all partitions of n with maximal element <= k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 4, 1, 0, 1, 4, 9, 6, 1, 0, 1, 5, 16, 19, 9, 1, 0, 1, 6, 25, 44, 39, 12, 1, 0, 1, 7, 36, 85, 116, 69, 16, 1, 0, 1, 8, 49, 146, 275, 260, 119, 20, 1, 0, 1, 9, 64, 231, 561, 751, 560, 189, 25, 1, 0
Offset: 0

Views

Author

Alois P. Heinz, Mar 20 2012

Keywords

Examples

			Square array A(n,k) begins:
  1,  1,   1,   1,   1,    1,    1, ...
  0,  1,   2,   3,   4,    5,    6, ...
  0,  1,   4,   9,  16,   25,   36, ...
  0,  1,   6,  19,  44,   85,  146, ...
  0,  1,   9,  39, 116,  275,  561, ...
  0,  1,  12,  69, 260,  751, 1812, ...
  0,  1,  16, 119, 560, 1955, 5552, ...
		

Crossrefs

Columns k=0-8 give: A000007, A000012, A002620(n+2), A038163, A054498, A181477, A181478, A181479, A181480.
Main diagonal gives: A209673.

Programs

  • Maple
    # First program:
    h:= (l, k)-> mul(mul((k+j-i)/(1+l[i] -j +add(`if`(l[t]>=j, 1, 0)
                     , t=i+1..nops(l))), j=1..l[i]), i=1..nops(l)):
    g:= proc(n, i, k, l)
          `if`(n=0, h(l, k), `if`(i<1, 0, g(n, i-1, k, l)+
          `if`(i>n, 0, g(n-i, i, k, [l[], i]))))
        end:
    A:= (n, k)-> `if`(n=0, 1, g(n, n, k, [])):
    seq(seq(A(n, d-n), n=0..d), d=0..12);
    # second program:
    gf:= k-> 1/((1-x)^k*(1-x^2)^(k*(k-1)/2)):
    A:= (n, k)-> coeff(series(gf(k), x, n+1), x, n):
    seq(seq(A(n, d-n), n=0..d), d=0..12);
  • Mathematica
    (* First program: *)
    h[l_, k_] := Product[Product[(k+j-i)/(1+l[[i]]-j + Sum[If[l[[t]] >= j, 1, 0], {t, i+1, Length[l]}]), {j, 1, l[[i]]}], {i, 1, Length[l]}]; g [n_, i_, k_, l_] := If[n == 0, h[l, k], If[i < 1, 0, g[n, i-1, k, l] + If[i > n, 0, g[n-i, i, k, Append[l, i]]]]]; a[n_, k_] := If[n == 0, 1, g[n, n, k, {}]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten
    (* second program: *)
    gf[k_] := 1/((1-x)^k*(1-x^2)^(k*(k-1)/2)); a[n_, k_] := Coefficient[Series[gf[k], {x, 0, n+1}], x, n]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Dec 09 2013, translated from Maple *)

Formula

G.f. of column k: 1/((1-x)^k*(1-x^2)^(k*(k-1)/2)).
A(n,k) = Sum_{i=0..k} C(k,i) * A138177(n,k-i). - Alois P. Heinz, Apr 06 2015

A139593 A139276(n) followed by A139272(n+1).

Original entry on oeis.org

0, 3, 11, 22, 38, 57, 81, 108, 140, 175, 215, 258, 306, 357, 413, 472, 536, 603, 675, 750, 830, 913, 1001, 1092, 1188, 1287, 1391, 1498, 1610, 1725, 1845, 1968, 2096, 2227, 2363, 2502, 2646, 2793, 2945, 3100, 3260, 3423, 3591, 3762
Offset: 0

Views

Author

Omar E. Pol, May 03 2008

Keywords

Comments

Sequence found by reading the line from 0, in the direction 0, 3, ... and the same line from 0, in the direction 0, 11, ..., in the square spiral whose vertices are the triangular numbers A000217.
A139593 appears (both numerically and via back of an envelope algebra, but not a publishable proof) to be the cumulative sum of A047470. - Markus J. Q. Roberts, Jul 12 2009

Examples

			Array begins:
   0,   3;
  11,  22;
  38,  57;
  81, 108;
		

Crossrefs

Programs

  • Mathematica
    LinearRecurrence[{2,0,-2,1},{0,3,11,22},50] (* Harvey P. Dale, Feb 09 2019 *)

Formula

Array read by rows: row n gives 8*n^2 + 3n, 8*(n+1)^2 - 5(n+1).
From Colin Barker, Sep 15 2013: (Start)
a(n) = (-1 + (-1)^n + 6*n + 8*n^2)/4.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
G.f.: -x*(5*x+3) / ((x-1)^3*(x+1)). (End)

Extensions

Edited by Omar E. Pol, Jul 13 2009
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