A092123 -id:A092123 - cp's OEIS frontend

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

Paul D. Hanna, Sep 18 2006

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

			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.

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.

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

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

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 *)

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 */

{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(""))

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.

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

A122893 Position of largest coefficient of n-th self-composition of (x+x^2) for n>=0.

Original entry on oeis.org

1, 1, 2, 5, 9, 19, 37, 74, 147, 294, 587, 1175, 2349, 4698, 9396, 18791, 37582

Offset: 0

Paul D. Hanna, Sep 19 2006

What is the limit a(n)/2^n = 0.573... ? Originated by Ralf Stephan in A092123 as the position of largest coefficient in the expansion of P(0)=x, P(n+1)=P(n)*[1+P(n)] (equivalent definition).

Cf. A092123, A122894; A122888.

P[n_] := P[n] = If[n < 1, x, P[n - 1]*(P[n - 1] + 1)]; Table[p = Expand[CoefficientList[P[n], x]]; Position[p, Max[p]][[1]][[1]] - 1, {n, 0, 12}] (* Vaclav Kotesovec, Nov 08 2018 *)

a(14)-a(15) from Vaclav Kotesovec, Nov 08 2018

a(16) from Vaclav Kotesovec, Nov 09 2018

A122894 Coefficient of x^(2^(n-1)) in the n-th self-composition of (x+x^2) for n>=1.

Original entry on oeis.org

1, 2, 9, 258, 293685, 531124770570, 2439717292075827330588969, 72554628124279239546273779187960042205300343234178

Offset: 1

Paul D. Hanna, Sep 19 2006

nonn

Originated by Ralf Stephan in A092123 as the 2^(n-1)th coefficient in the expansion of P(0)=x, P(n+1)=P(n)*[1+P(n)] (equivalent definition). Next term is too large to include.

			a(1) = 1 = [x^1] (x + x^2).
a(2) = 2 = [x^2] (x + 2*x^2 + 2*x^3 + x^4).
a(3) = 9 = [x^4] (x + 3*x^2 + 6*x^3 + 9*x^4 + 10*x^5 + 8*x^6 + 4*x^7 + x^8).

Cf. A092123, A122893; A122888.

{a(n)=local(F=x+x^2, G=x+x*O(x^(2^(n-1)))); if(n<1, 0, for(i=1, n, G=subst(F, x, G)); return(polcoeff(G, 2^(n-1), x)))}

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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.

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A122893 Position of largest coefficient of n-th self-composition of (x+x^2) for n>=0.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Extensions

A122894 Coefficient of x^(2^(n-1)) in the n-th self-composition of (x+x^2) for n>=1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI