A126348 -id:A126348 - cp's OEIS frontend

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).

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

A307599 Expansion of Product_{k>=1} (1 - x^k/(1 - x)).

Original entry on oeis.org

1, -1, -2, -2, -1, 2, 6, 11, 15, 16, 11, -2, -26, -61, -105, -152, -192, -209, -183, -89, 98, 400, 830, 1385, 2035, 2715, 3314, 3668, 3556, 2703, 790, -2521, -7550, -14542, -23591, -34546, -46901, -59670, -71261, -79358, -80830, -71690, -47133, -1684, 70504, 175168, 317232

Offset: 0

Seiichi Manyama, Apr 17 2019

sign

Seiichi Manyama, Table of n, a(n) for n = 0..500

Convolution inverse of A227682.

Cf. A126348, A307601, A307602.

m = 46; CoefficientList[Series[Product[1 - x^k/(1 - x), {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, May 14 2021 *)

N=66; x='x+O('x^N); Vec(prod(k=1, N, 1-x^k/(1-x)))

N=66; x='x+O('x^N); Vec(exp(-sum(k=1, N, x^k*sumdiv(k, d, 1/(d*(1-x)^d)))))

G.f.: exp( - Sum_{k>=1} x^k * Sum_{d|k} 1/(d*(1-x)^d)).

A307601 Expansion of Product_{k>=1} (1 - x^k/(1 + x)).

Original entry on oeis.org

1, -1, 0, 0, -1, 2, -4, 7, -11, 16, -21, 26, -30, 33, -33, 28, -14, -13, 59, -131, 238, -390, 598, -873, 1225, -1663, 2194, -2822, 3544, -4347, 5202, -6059, 6838, -7420, 7633, -7238, 5911, -3226, -1365, 8552, -19190, 34320, -55189, 83266, -120254, 168094, -228958

Offset: 0

Seiichi Manyama, Apr 18 2019

Seiichi Manyama, Table of n, a(n) for n = 0..500

Cf. A126348, A307599, A307602.

m = 46; CoefficientList[Series[Product[1 - x^k/(1 + x), {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, May 14 2021 *)

N=66; x='x+O('x^N); Vec(prod(k=1, N, 1-x^k/(1+x)))

N=66; x='x+O('x^N); Vec(exp(-sum(k=1, N, x^k*sumdiv(k, d, 1/(d*(1+x)^d)))))

G.f.: exp(-Sum_{k>=1} x^k * Sum_{d|k} 1/(d*(1+x)^d)).

A307602 Expansion of Product_{k>=1} (1 + x^k/(1 + x)).

Original entry on oeis.org

1, 1, 0, 2, -1, 4, -2, 5, -1, 4, 3, -2, 16, -21, 47, -62, 104, -131, 191, -229, 304, -344, 420, -437, 477, -413, 336, -76, -270, 927, -1792, 3155, -4904, 7402, -10519, 14694, -19761, 26226, -33847, 43162, -53776, 66178, -79679, 94562, -109606, 124618, -137468, 147061

Offset: 0

Seiichi Manyama, Apr 18 2019

sign

Seiichi Manyama, Table of n, a(n) for n = 0..500

Cf. A126348, A307599, A307601.

nmax = 50; CoefficientList[Series[Product[(1 + x + x^k)/(1 + x), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 16 2019 *)

N=66; x='x+O('x^N); Vec(prod(k=1, N, 1+x^k/(1+x)))

N=66; x='x+O('x^N); Vec(exp(sum(k=1, N, x^k*sumdiv(k, d, (-1)^(d+1)/(d*(1+x)^d)))))

G.f.: exp(Sum_{k>=1} x^k * Sum_{d|k} (-1)^(d+1) / (d*(1+x)^d)).

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

Original entry on oeis.org

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]}

A126349 Largest term in rows of triangle A126347, in which row sums equal Bell numbers (A000110).

Original entry on oeis.org

1, 1, 1, 2, 4, 10, 29, 101, 392, 1721, 8180, 42234, 232893, 1376660, 8608564, 56880860, 395780193, 2887365769, 22022011448, 175148070545, 1451626482840, 12489021006783, 111407399288300, 1028527130941484, 9818171726308337

Offset: 0

Paul D. Hanna, Dec 31 2006

nonn

In triangle A126347, 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.

Cf. A126347, A126348.

PARI

A253830 Triangular array with g.f. Product_{n >= 1} (1 + (x*z)^n/(1 - z)).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 1, 3, 2, 0, 1, 1, 4, 3, 3, 0, 1, 1, 5, 4, 5, 4, 0, 1, 1, 6, 5, 7, 8, 4, 0, 1, 1, 7, 6, 9, 13, 10, 6, 0, 1, 1, 8, 7, 11, 19, 16, 13, 8, 0, 1, 1, 9, 8, 13, 26, 23, 22, 18, 10, 0, 1, 1, 10, 9, 15, 34, 31, 33, 31, 25, 12, 0, 1, 1, 11, 10, 17, 43, 40, 46, 47, 47, 30, 15

Offset: 0

Peter Bala, Jan 20 2015

A colored composition of n is defined as a composition of n where each part p comes in one of p colors (denoted by an integer from 1 to p) and the color numbers are nondecreasing through the composition.
The color numbers thus form a partition, called the color partition, of some integer. For example, 2(c1) + 1(c1) + 5(c3) + 4(c3) + 6(c4) is a colored composition of 18 (the color number of a part is shown after the part prefaced by the letter c) and has the associated color partition (1,1,3,3,4).
T(n,k) equals the number of colored compositions of n whose associated color partition has distinct parts with sum (called the weight of the color partition) equal to k. An example is given below.

			Triangle begins
n\k| 0  1  2  3  4  5  6  7
= = = = = = = = = = = = = =
0  | 1
1  | 0  1
2  | 0  1  1
3  | 0  1  1  2
4  | 0  1  1  3  2
5  | 0  1  1  4  3  3
6  | 0  1  1  5  4  5  4
7  | 0  1  1  6  5  7  8  4
...
Row 5 polynomial: x + x^2 + 4*x^3 + 3*x*4 + 3*x^5.
Colored             x^(weight of color partition)
compositions
of 5 with
distinct colored
parts
= = = = = = = = = = = = = = = = = = = = = =
5(c1)                        x
5(c2)                        x^2
1(c1) + 4(c2)                x^3
2(c1) + 3(c2)                x^3
3(c1) + 2(c2)                x^3
5(c3)                        x^3
1(c1) + 4(c3)                x^4
2(c1) + 3(c3)                x^4
5(c4)                        x^4
1(c1) + 4(c4)                x^5
2(c2) + 3(c3)                x^5
5(c5)                        x^5

P. Bala, Colored Compositions

Cf. A008289, A126348 (row sums), A253829.

G := product(1+(x*z)^j/(1-z), j = 1 .. 12): Gser := simplify(series(G, z = 0, 14)): for n to 12 do P[n] := coeff(Gser, z^n) end do: for n to 12 do seq(coeff(P[n], x^j), j = 1 .. n) end do;

G.f.: G(x,z) := Product_{n >= 1} (1 + (x*z)^n/(1 - z)) = 1 + x*z + (x + x^2)*z^2 + (x + x^2 + 2*x^3)*z^3 + (x + x^2 + 3*x^3 + 2*x^4)*z^4 + .... Note, G(x*z/(x - 1),(x - 1)/x) is the generating function of A008289.
T(n,k) = Sum_{i = 1..k} binomial(i+n-k-1,i-1)*A008289(k,i).
Row sums are A126348.

A336989 Expansion of Product_{k>=1} (1 + x^k / (1 - k*x)).

Original entry on oeis.org

1, 1, 2, 5, 13, 36, 107, 343, 1184, 4391, 17448, 74082, 335131, 1610301, 8191728, 43973853, 248305235, 1470474074, 9107950029, 58856529464, 395914407606, 2766669954699, 20047716439541, 150384068021507, 1166037568730402, 9332538119883810, 77004693701288392, 654279226353488820

Offset: 0

Seiichi Manyama, Aug 10 2020

nonn

Cf. A126348, A336980, A336990, A336991.

m = 27; CoefficientList[Series[Product[1 + x^k/(1 - k*x), {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, Aug 10 2020 *)

N=40; x='x+O('x^N); Vec(prod(k=1, N, 1+x^k/(1-k*x)))

N=40; x='x+O('x^N); Vec(exp(sum(k=1, N, x^k*sumdiv(k, d, (-1)^(d+1)/(d*(1-k/d*x)^d)))))

G.f.: exp(Sum_{k>=1} x^k * Sum_{d|k} (-1)^(d+1) / (d * (1 - k/d * x)^d)).

A307761 L.g.f.: log(Product_{k>=1} (1 + x^k/(1 - x))) = Sum_{k>=1} a(k)*x^k/k.

Original entry on oeis.org

1, 3, 7, 11, 16, 21, 29, 35, 43, 48, 56, 65, 79, 87, 97, 99, 103, 111, 134, 156, 182, 190, 185, 161, 141, 133, 178, 263, 378, 471, 497, 387, 161, -133, -341, -313, 75, 782, 1645, 2300, 2379, 1596, -42, -2222, -4232, -5241, -4464, -1551, 3263, 9023, 14287, 17249, 16219, 9912, -2074

Offset: 1

Seiichi Manyama, Apr 27 2019

sign

			L.g.f.: L(x) = x/1 + 3*x^2/2 + 7*x^3/3 + 11*x^4/4 + 16*x^5/5 + 21*x^6/6 + 29*x^7/7 + 35*x^8/8 + ... .
exp(L(x)) = 1 + x + 2*x^2 + 4*x^3 + 7*x^4 + 12*x^5 + 20*x^6 + 33*x^7 + 53*x^8 + ... + A126348(n)*x^n + ... .

Cf. A126348, A307674, A307675, A307762.

N=66; x='x+O('x^N); Vec(x*deriv(log(prod(k=1, N, 1+x^k/(1-x)))))

N=66; x='x+O('x^N); Vec(x*deriv(sum(k=1, N, x^k*sumdiv(k, d, (-1)^(d+1)/(d*(1-x)^d)))))

Product {k>=1} (1 + x^k/(1 - x)) = exp(Sum_{k>=1} a(k)*x^k/k).

A322613 Expansion of e.g.f. Product_{k>=1} (1 - log(1 - x)*x^k).

Original entry on oeis.org

1, 0, 2, 9, 44, 370, 3084, 32088, 336384, 4407408, 59113440, 896773680, 14403234240, 250498939392, 4625127900288, 92232410538240, 1925532322237440, 42709138254167040, 997150775080043520, 24416143271431649280, 626110124433676185600, 16824255461119247339520, 471015493365385119191040

Offset: 0

Ilya Gutkovskiy, Dec 20 2018

nonn

Cf. A126348, A265952, A322612.

seq(coeff(series(factorial(n)*mul((1-log(1-x)*x^k),k=1..n),x,n+1), x, n), n = 0 .. 22); # Muniru A Asiru, Dec 21 2018

nmax = 22; CoefficientList[Series[Product[(1 - Log[1 - x] x^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[Sum[Sum[(-1)^(d + 1) Log[1/(1 - x)]^d/d, {d, Divisors[k]}] x^k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!

E.g.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(d+1)*log(1/(1 - x))^d/d ) * x^k).

cp's OEIS Frontend

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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A307599 Expansion of Product_{k>=1} (1 - x^k/(1 - x)).

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Mathematica

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A307601 Expansion of Product_{k>=1} (1 - x^k/(1 + x)).

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Mathematica

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A307602 Expansion of Product_{k>=1} (1 + x^k/(1 + x)).

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Mathematica

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

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A126349 Largest term in rows of triangle A126347, in which row sums equal Bell numbers (A000110).

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PARI

A253830 Triangular array with g.f. Product_{n >= 1} (1 + (x*z)^n/(1 - z)).

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Maple

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A336989 Expansion of Product_{k>=1} (1 + x^k / (1 - k*x)).

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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).