A126470 -id:A126470 - cp's OEIS frontend

A126471 Limit of reversed rows of triangle A126470, in which row sums equal the factorials.

1, 1, 3, 5, 12, 17, 39, 58, 108, 170, 310, 449, 791, 1181, 1960, 2915, 4668, 6822, 10842, 15818, 24254, 35061, 53213, 76061, 113822, 162631, 238660, 337764, 491319, 690530, 994390, 1391968, 1982724, 2757196, 3896450, 5382342, 7546547, 10384787

Offset: 0

Paul D. Hanna, Dec 31 2006

nonn

In triangle A126470, row n lists coefficients of q in F(n,q) that satisfies: F(n,q) = Sum_{k=0..n-1} C(n-1,k)*F(k,q)*F(n-k-1,q)*q^k for n>0, with F(0,q) = 1.

			Row functions F(n,q) of triangle A126470 begin:
F(0,q) = F(1,q) = 1;
F(1,q) = 1 + q;
F(2,q) = 1 + 3*q + q^2 + q^3;
F(3,q) = 1 + 6*q + 7*q^2 + 5*q^3 + 3*q^4 + q^5 + q^6.

Cf. A126470, A126472; Bell number variant: A126348.

{F(n,q)=if(n==0,1,sum(k=0,n-1,binomial(n-1,k)*F(k,q)*F(n-k-1,q)*q^k))} {a(n)=Vec(F(n+1,q)+O(q^(n*(n-1)/2+1)))[n*(n-1)/2+1]}

A126472 Largest term in rows of triangle A126470, in which row sums equal factorials.

Original entry on oeis.org

1, 1, 1, 3, 7, 26, 136, 784, 5189, 40639, 357638, 3472240, 37835023, 449901885, 5763035223, 79697811456, 1185091117180, 18789393725696, 316557520950918, 5664877511610114, 106815500916218123, 2123415404236087790

Offset: 0

Paul D. Hanna, Dec 31 2006

nonn

In triangle A126470, row n lists coefficients of q in F(n,q) that satisfies: F(n,q) = Sum_{k=0..n-1} C(n-1,k)*F(k,q)*F(n-k-1,q)*q^k for n>0, with F(0,q) = 1.

			Row functions F(n,q) of triangle A126470 begin:
F(0,q) = F(1,q) = 1;
F(1,q) = 1 + q;
F(2,q) = 1 + 3*q + q^2 + q^3;
F(3,q) = 1 + 6*q + 7*q^2 + 5*q^3 + 3*q^4 + q^5 + q^6.

Cf. A126470, A126471.

PARI

A126347 Triangle, read by rows, where row n lists coefficients of q in B(n,q) that satisfies: B(n,q) = Sum_{k=0..n-1} C(n-1,k)B(k,q)q^k for n>0, with B(0,q) = 1; row sums equal the Bell numbers: B(n,1) = A000110(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 4, 2, 1, 1, 1, 4, 6, 10, 9, 7, 7, 4, 2, 1, 1, 1, 5, 10, 20, 25, 26, 29, 26, 20, 14, 12, 7, 4, 2, 1, 1, 1, 6, 15, 35, 55, 71, 90, 101, 100, 89, 82, 68, 53, 38, 26, 20, 12, 7, 4, 2, 1, 1, 1, 7, 21, 56, 105, 161, 231, 302, 356, 379, 392, 384, 358, 314, 262

Offset: 0

Paul D. Hanna, Dec 31 2006, May 28 2007

Limit of reversed rows equals A126348. Largest term in rows equal A126349.

			Number of terms in row n is: n*(n-1)/2 + 1.
Row functions B(n,q) begin:
  B(0,q) = 1;
  B(1,q) = 1;
  B(2,q) = 1 + q;
  B(3,q) = 1 + 2*q + q^2 + q^3;
  B(4,q) = 1 + 3*q + 3*q^2 + 4*q^3 + 2*q^4 + q^5 + q^6.
Triangle begins:
  1;
  1;
  1, 1;
  1, 2, 1, 1;
  1, 3, 3, 4, 2, 1, 1;
  1, 4, 6, 10, 9, 7, 7, 4, 2, 1, 1;
  1, 5, 10, 20, 25, 26, 29, 26, 20, 14, 12, 7, 4, 2, 1, 1;
  1, 6, 15, 35, 55, 71, 90, 101, 100, 89, 82, 68, 53, 38, 26, 20, 12, 7, 4, 2, 1, 1;
  ...

Alois P. Heinz, Rows n = 0..60, flattened
Carl G. Wagner, Partition Statistics and q-Bell Numbers (q = -1), J. Integer Seqs., Vol. 7, 2004.

Row sums give A000110.
Cf. A126348, A126349; factorial variant: A126470.
Cf. A346772.

b:= proc(n, m, t) option remember; `if`(n=0, x^t,
      add(b(n-1, max(m, j), t+j) , j=1..m+1))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=n..degree(p)))(b(n, 0$2)):
seq(T(n), n=0..8);  # Alois P. Heinz, Aug 02 2021

B[0, ] = 1; B[n, q_] := B[n, q] = Sum[Binomial[n-1, k] B[k, q] q^k, {k, 0, n-1}] // Expand; Table[CoefficientList[B[n, q], q], {n, 0, 8}] // Flatten (* Jean-François Alcover, Nov 08 2016 *)

{B(n,q)=if(n==0,1,sum(k=0,n-1,binomial(n-1,k)*B(k,q)*q^k))}
row(n)={Vec(B(n, 'q)+O('q^(n*(n-1)/2+1)))}

/* Alternative formula for the n-th q-Bell number (row n): */ {B(n,q)=local(inf=100);round((0^n + sum(k=1, inf,((q^k-1)/(q-1))^n/prod(i=1,k,(q^i-1)/(q-1)))) / prod(k=1, inf,1 + (q-1)/q^k))}

G.f. for row n: B(n,q) = 1/E_q*{0^n + Sum_{k>=1} [(q^k-1)/(q-1)]^n / q-Factorial(k)}, where q-Factorial(k) = Product_{j=1..k} [(q^j-1)/(q-1)] and where E_q = Sum_{n>=0} 1/q-Factorial(n) = Product_{n>=1} (1+(q-1)/q^n).
Sum_{k=0..n*(n-1)/2} (n+k) * T(n,k) = A346772(n). - Alois P. Heinz, Aug 02 2021
Conjecture: R(n,n) is the (n+1)-th reversed row polynomial where R(0,0) = 1, R(n,k) = R(n-1,n-1) + x^n * Sum_{j=0..k-1} R(n-1,j) for 0 <= k <= n. - Mikhail Kurkov, Jul 06 2025

Keyword:tabl changed to tabf by R. J. Mathar, Oct 21 2010

A126444 a(n) = Sum_{k=0..n-1} C(n-1,k)a(k)a(n-1-k)*2^k for n>0, with a(0)=1.

Original entry on oeis.org

1, 1, 3, 19, 225, 4801, 185523, 13298659, 1815718305, 481790947681, 251592291767043, 260427247041910099, 536497603929547755585, 2204489516030261302702561, 18090090482887693483393912563, 296659627048147988400872084439139

Offset: 0

Paul D. Hanna, Jan 01 2007

nonn

Generated by a generalization of a recurrence for the factorials.

Vincenzo Librandi, Table of n, a(n) for n = 0..80

Cf. A126470.

b = ConstantArray[0,21]; b[[1]]=1; b[[2]]=1; Do[b[[n+1]] = Sum[Binomial[n-1,k]*b[[k+1]]*b[[n-k]]*2^k,{k,0,n-1}],{n,2,20}]; b  (* Vaclav Kotesovec, Feb 23 2014 *)

a(n)=if(n==0,1,sum(k=0,n-1,binomial(n-1,k)*a(k)*a(n-1-k)*2^k))

{a(n)=local(A=1+x);for(i=0,n,A=1+intformal(A*subst(A,x,2*x+x*O(x^n))));n!*polcoeff(A,n,x)} \\ Paul D. Hanna, Nov 22 2008

a(n) = Sum_{k=0..n*(n-1)/2} A126470(n,k)*2^k.
E.g.f. satisfies: A'(x) = A(x)*A(2x) with A(0)=1; the logarithmic derivative of e.g.f. A(x) equals A(2x). - Paul D. Hanna, Nov 22 2008
a(n) ~ c * 2^(n*(n-1)/2), where c = 7.32081762965209017732559... - Vaclav Kotesovec, Feb 23 2014

More terms from Vincenzo Librandi, Feb 25 2014

A232433 E.g.f. satisfies: A(x,q) = exp( Integral A(x,q)A(qx,q) dx ).

Original entry on oeis.org

1, 1, 2, 1, 6, 6, 2, 1, 24, 36, 22, 14, 6, 2, 1, 120, 240, 210, 160, 104, 56, 32, 14, 6, 2, 1, 720, 1800, 2040, 1830, 1448, 992, 674, 408, 232, 128, 68, 32, 14, 6, 2, 1, 5040, 15120, 21000, 21840, 19824, 15834, 12144, 8758, 5904, 3860, 2442, 1482, 870, 492, 260, 142, 68, 32, 14, 6, 2, 1

Offset: 0

Paul D. Hanna, Nov 23 2013

			E.g.f.: A(x,q) = 1 + (1)*x + (2 + q)*x^2/2! + (6 + 6*q + 2*q^2 + q^3)*x^3/3!
+ (24 + 36*q + 22*q^2 + 14*q^3 + 6*q^4 + 2*q^5 + q^6)*x^4/4!
+ (120 + 240*q + 210*q^2 + 160*q^3 + 104*q^4 + 56*q^5 + 32*q^6 + 14*q^7 + 6*q^8 + 2*q^9 + q^10)*x^5/5! +...
The triangle of coefficients T(n,k) of x^n*q^k, for n>=0, k=0..n*(n-1)/2, in e.g.f. A(x,q) begins:
[1];
[1];
[2, 1];
[6, 6, 2, 1];
[24, 36, 22, 14, 6, 2, 1];
[120, 240, 210, 160, 104, 56, 32, 14, 6, 2, 1];
[720, 1800, 2040, 1830, 1448, 992, 674, 408, 232, 128, 68, 32, 14, 6, 2, 1];
[5040, 15120, 21000, 21840, 19824, 15834, 12144, 8758, 5904, 3860, 2442, 1482, 870, 492, 260, 142, 68, 32, 14, 6, 2, 1];
[40320, 141120, 231840, 275520, 280056, 251496, 212112, 170424, 129716, 95248, 67632, 46616, 31280, 20576, 13142, 8232, 5004, 2954, 1706, 966, 524, 276, 142, 68, 32, 14, 6, 2, 1]; ...
The limit of the reversed rows (A232434) begins:
[1, 2, 6, 14, 32, 68, 142, 276, 542, 1022, 1876, 3394, 6066, 10628, ...].

Paul D. Hanna, Table of n, a(n) for n = 0..9919; rows 0..39 in flattened triangle.
Miguel A. Mendez, Combinatorial differential operators in: Faà di Bruno formula, enumeration of ballot paths, enriched rooted trees and increasing rooted trees, arXiv:1610.03602 [math.CO], 2016.

Cf. A232434, A126470, A001286, A075183.

nmax = 8; A[, ] = 0; Do[A[x_, q_] = Exp[Integrate[A[x, q] A[q x, q], x]] + O[x]^n // Normal // Simplify, {n, nmax}];
CoefficientList[#, q]& /@ (CoefficientList[A[x, q], x] Range[0, nmax-1]!) // Flatten (* Jean-François Alcover, Oct 27 2018 *)

{T(n,k)=local(A=1+x);for(i=1,n,A=exp(intformal(A*subst(A,x,x*y +x*O(x^n)),x)));n!*polcoeff(polcoeff(A,n,x),k,y)}
for(n=0,12,for(k=0,n*(n-1)/2,print1(T(n,k),", "));print(""))

E.g.f. satisfies: d/dx A(x,q) = A(x,q)^2 * A(q*x,q).
Row sums equal the odd double factorials.
Limit of reversed rows yield A232434.

cp's OEIS Frontend

A126471 Limit of reversed rows of triangle A126470, in which row sums equal the factorials.

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A126472 Largest term in rows of triangle A126470, in which row sums equal factorials.

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PARI

A126347 Triangle, read by rows, where row n lists coefficients of q in B(n,q) that satisfies: B(n,q) = Sum_{k=0..n-1} C(n-1,k)*B(k,q)*q^k for n>0, with B(0,q) = 1; row sums equal the Bell numbers: B(n,1) = A000110(n).

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A126444 a(n) = Sum_{k=0..n-1} C(n-1,k)*a(k)*a(n-1-k)*2^k for n>0, with a(0)=1.

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Mathematica

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A232433 E.g.f. satisfies: A(x,q) = exp( Integral A(x,q)*A(q*x,q) dx ).

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Mathematica

PARI

Formula

A126347 Triangle, read by rows, where row n lists coefficients of q in B(n,q) that satisfies: B(n,q) = Sum_{k=0..n-1} C(n-1,k)B(k,q)q^k for n>0, with B(0,q) = 1; row sums equal the Bell numbers: B(n,1) = A000110(n).

A126444 a(n) = Sum_{k=0..n-1} C(n-1,k)a(k)a(n-1-k)*2^k for n>0, with a(0)=1.

A232433 E.g.f. satisfies: A(x,q) = exp( Integral A(x,q)A(qx,q) dx ).