A122888 Triangle, read by rows, where row n lists the coefficients of x^k, k=1..2^n, in the n-th iteration of (x + x^2) for n>=0.

1, 1, 1, 1, 2, 2, 1, 1, 3, 6, 9, 10, 8, 4, 1, 1, 4, 12, 30, 64, 118, 188, 258, 302, 298, 244, 162, 84, 32, 8, 1, 1, 5, 20, 70, 220, 630, 1656, 4014, 8994, 18654, 35832, 63750, 105024, 160120, 225696, 293685, 352074, 387820, 391232, 359992, 300664, 226580

Offset: 0

Author

Paul D. Hanna, Sep 18 2006

Keywords

• nonn
• tabf

Comments

T(n, k) is the number of strings of length k-1 on the alphabet {1, 2, ..., n} such that between every two occurrences of a letter i there is an occurrence of a letter strictly larger than i. For example, for n = 3, k = 4 we have the strings 121, 131, 232 and the six permutations of 123. - Joel B. Lewis, May 06 2008

Examples

			Triangle begins:
  1;
  1, 1;
  1, 2, 2, 1;
  1, 3, 6, 9, 10, 8, 4, 1;
  1, 4, 12, 30, 64, 118, 188, 258, 302, 298, 244, 162, 84, 32, 8, 1;
  1, 5, 20, 70, 220, 630, 1656, 4014, 8994, 18654, 35832, 63750,...;
  1, 6, 30, 135, 560, 2170, 7916, 27326, 89582, 279622, 832680,...;
  1, 7, 42, 231, 1190, 5810, 27076, 121023, 520626, 2161158,...;
  1, 8, 56, 364, 2240, 13188, 74760, 409836, 2179556, 11271436,...;
  1, 9, 72, 540, 3864, 26628, 177744, 1153740, 7303164, 45179508,...;
  1, 10, 90, 765, 6240, 49260, 378312, 2836548, 20817588,...; ...
Multiplying the g.f. of column k by (1-x)^k, k>=1, with leading zeros,
 yields the g.f. of row k in the triangle A122890:
  1;
  0, 1;
  0, 0, 2;
  0, 0, 1, 5;
  0, 0, 0, 10, 14;
  0, 0, 0, 8, 70, 42;
  0, 0, 0, 4, 160, 424, 132;
  0, 0, 0, 1, 250, 1978, 2382, 429;
  0, 0, 0, 0, 302, 6276, 19508, 12804, 1430; ...
in which the main diagonal is the Catalan numbers,
 and the row sums form the factorials.
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..2046 (rows 0..10)
• Art of Problem Solving forum, Strings on [n] with certain restrictions.
• Eunmi Choi, Obtuse matrix of arithmetic table, East Asian Math. J. (2024) Vol. 40, No. 3, 329-339. See p. 11.

Crossrefs

Cf. A007018 (row sums), diagonals: A112317, A112319, A122887; A092123 (largest term in row); A122889 (antidiagonal sums); A122890 (related triangle).

Programs

Maple

b:= proc(n) option remember; `if`(n=0, x,
      expand((x-> x+x^2)(b(n-1))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n)):
seq(T(n), n=0..5);  # Alois P. Heinz, Mar 14 2016

Mathematica

f[0][x_] = x; f[n_][x_] := f[n][x] = f[n-1][x+x^2]; row[n_] := CoefficientList[f[n][x], x] // Rest; Table[row[n], {n, 0, 5} ] // Flatten (* Jean-François Alcover, Sep 10 2012 *)

Maxima

T(m,n):=if m=0 and n=1 then 1 else if m=0 and n>1 then 0 else  if m=1 then binomial(1,n-1) else sum(binomial(i,n-i)*T(m-1,i),i,1,n); /* Vladimir Kruchinin, May 19 2012 */

PARI

{T(n,k)=local(F=x+x^2, G=x+x*O(x^k)); if(n<0, 0, for(i=1, n, G=subst(F, x, G)); return(polcoeff(G, k, x)))}
for(n=0, 6, for(k=1, 2^n, print1(T(n, k), ", ")); print(""))

Formula

T(n,k) = [x^k] F_n(x) where F_{n+1}(x) = F_n(x+x^2) for n>=1, with F_0(x)=x.

Extensions

Name changed slightly by Paul D. Hanna, Apr 29 2013