A179525 -id:A179525 - cp's OEIS frontend

A186737 G.f. satisfies: A(x) = Sum_{n>=0} Product_{k=1..n} ((1 + x*A(x))^k - 1).

1, 1, 3, 14, 82, 563, 4390, 38273, 370090, 3951524, 46436236, 598606795, 8431113021, 129134793211, 2139840753676, 38161937079832, 728817179712481, 14837170081079164, 320656534559643735, 7329984468062242588, 176662137501934534082, 4476371962949911027001, 118946746110478235868525

Offset: 0

Paul D. Hanna, Apr 23 2013

nonn

This is an example of Peter Bala's identity (cf. A158690):

Sum_{n>=0} Product_{k=1..n} (q^k - 1) = Sum_{n>=0} q^(-n^2) * Product_{k=1..n} (q^(2*k-1) - 1) at q = 1+x*A(x).

			G.f.: A(X) = 1 + x + 3*x^2 + 14*x^3 + 82*x^4 + 563*x^5 + 4390*x^6 +...
where
A(x) = 1 + ((1+x*A(x))-1) + ((1+x*A(x))-1)*((1+x*A(x))^2-1) + ((1+x*A(x))-1)*((1+x*A(x))^2-1)*((1+x*A(x))^3-1) + ((1+x*A(x))-1)*((1+x*A(x))^2-1)*((1+x*A(x))^3-1)*((1+x*A(x))^4-1) +...
Let q = 1+x*A(x), then:
A(x) = 1 + (q-1)/q + (q-1)*(q^3-1)/q^4 + (q-1)*(q^3-1)*(q^5-1)/q^9 + (q-1)*(q^3-1)*(q^5-1)*(q^7-1)/q^16 + (q-1)*(q^3-1)*(q^5-1)*(q^7-1)*(q^9-1)/q^25 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..115
Hsien-Kuei Hwang, and Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], p. 33, 2019.

Cf. A179525.

nmax = 22; sol = {a[0] -> 1};
Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x] - Sum[Product[ ((1 + x A[x])^k - 1), {k, 1, j}], {j, 0, n}] + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];
sol /. Rule -> Set;
a /@ Range[0, nmax] (* Jean-François Alcover, Nov 02 2019 *)

{a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,n,prod(k=1,m,(1+x*A+x*O(x^n))^k - 1)));polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))

{a(n)=local(A=1+x, q); for(i=1,n,q=(1+x*A +x*O(x^n)); A=sum(m=0, n, q^(-m^2)*prod(k=1, m, (q^(2*k-1)-1)))); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

G.f. satisfies:
(1) A(x) = Sum_{n>=0} (1+x*A(x))^(-n^2) * Product_{k=1..n} ((1+x*A(x))^(2*k-1) - 1).
(2) A(x) = 1/(1 - (q-1)/(q - (q^2-1)/(q^2 - (q^3-1)/(q^3 - (q^4-1)/(q^4 - (q^5-1)/(q^5 -...)))))), a continued fraction, where q = 1+x*A(x).
a(n) ~ 6*sqrt(2) * exp(Pi^2/24) * 12^n * n! / Pi^(2*n+2). - Vaclav Kotesovec, May 05 2014

A207434 L.g.f.: log( Sum_{n>=0} Product_{k=1..n} ((1+x)^k - 1) ) = Sum_{n>=1} a(n)*x^n/n.

Original entry on oeis.org

1, 3, 16, 103, 796, 7104, 71807, 810239, 10095145, 137686648, 2040943180, 32679948256, 562281127266, 10347659040127, 202849692259846, 4220573966037231, 92900793975348826, 2156973952747274733, 52686155932369860221, 1350605860832381895768, 36256679580764579284889

Offset: 1

Paul D. Hanna, Feb 19 2012

nonn

			L.g.f.: L(x) = x + 3*x^2/2 + 16*x^3/3 + 103*x^4/4 + 796*x^5/5 + 7104*x^6/6 + ...
where exponentiation yields the g.f. of A179525:
exp(L(x)) = 1 + x + 2*x^2 + 7*x^3 + 33*x^4 + 197*x^5 + 1419*x^6 + 11966*x^7 + ...
such that, by definition,
exp(L(x)) = 1 + ((1+x)-1) + ((1+x)-1)*((1+x)^2-1) + ((1+x)-1)*((1+x)^2-1)*((1+x)^3-1) + ...

G. C. Greubel, Table of n, a(n) for n = 1..250
Hsien-Kuei Hwang, Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019.

Cf. A179525 (exp).

Rest@With[{m = 25}, CoefficientList[Series[Log[Sum[Product[(1+x)^k -1, {k, j}], {j,0,m+2}]], {x,0,m}], x]*Range[0, m]] (* G. C. Greubel, Feb 05 2020 *)

{a(n)=n*polcoeff(log(sum(m=0, n, prod(k=1, m, (1+x)^k-1, 1+x*O(x^n)))), n)}
for(n=1,31,print1(a(n),","))

L.g.f.: log( Sum_{n>=0} 1/(1+x)^(n^2) * Product_{k=1..n} ((1+x)^(2*k-1) - 1) ).

A207649 E.g.f.: Sum_{n>=0} 1/n! * Product_{k=1..n} ((1+x)^k - 1).

Original entry on oeis.org

1, 1, 2, 9, 60, 580, 7380, 119700, 2387280, 57211056, 1615703040, 52950688560, 1988213895360, 84622184412480, 4045503794492160, 215531680987022400, 12709328457015590400, 824497514996060505600, 58531734588587835294720, 4525412571041289046479360, 379428698362674642409728000, 34365923697307101211784908800, 3350637193653655199937586944000

Offset: 0

Paul D. Hanna, Feb 19 2012

nonn

			E.g.f.: A(x) = 1 + x + 2*x^2/2! + 9*x^3/3! + 60*x^4/4! + 580*x^5/5! +...
such that, by definition,
A(x) = 1 + ((1+x)-1) + ((1+x)-1)*((1+x)^2-1)/2! + ((1+x)-1)*((1+x)^2-1)*((1+x)^3-1)/3! + ((1+x)-1)*((1+x)^2-1)*((1+x)^3-1)*((1+x)^4-1)/4! +...
or, more explicitly,
A(x) = 1 + x + x^2*(2+x)/2! + x^3*(2+x)*(3+3*x+x^2)/3! + x^4*(2+x)*(3+3*x+x^2)*(4+6*x+4*x^2+x^3)/4! + x^5*(2+x)*(3+3*x+x^2)*(4+6*x+4*x^2+x^3)*(5+10*x+10*x^2+5*x^3+x^4)/5! +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..300

Cf. A179525.

{a(n)=n!*polcoeff(sum(m=0,n,1/m!*prod(k=1,m,(1+x)^k-1 +x*O(x^n)) ),n)}
for(n=0,25,print1(a(n),", "))

a(21)-a(22) corrected by Andrew Howroyd, Feb 22 2018

A289316 The number of upper-triangular matrices whose nonzero entries are positive odd numbers summing to n and each row contains a nonzero entry.

Original entry on oeis.org

1, 1, 2, 8, 37, 219, 1557, 12994, 124427, 1344506, 16178891, 214522339, 3107144562, 48805300668, 826268787588, 14998055299920, 290550119360174, 5983278021430064, 130512410617529321, 3006012061455129053, 72900477505718600661

Offset: 0

Peter Bala, Jul 24 2017

A row-Fishburn matrix of size n is defined to be an upper-triangular matrix with nonnegative integer entries which sum to n and each row contains a nonzero entry. See A158691. Here we are considering row-Fishburn matrices where the nonzero entries are all odd.

The g.f. F(x) for primitive row_Fishburn matrices (i.e., row_Fishburn matrices with entries restricted to the set {0,1}), is F(x) = Sum_{n>=0} Product_{k=1..n} ( (1 + x)^k - 1 ). See A179525. Let C(x) = x/(1 - x^2) = x + x^3 + x^5 + x^7 + .... Then appplying Lemma 2.2.22 of Goulden and Jackson gives the g.f. for the present sequence as the composition F(C(x)).

			a(3) = 8: The eight row-Fishburn matrices of size 3 with odd nonzero entries are
(3) /1 1\
    \0 1/
/1 0 0\  /0 1 0\  /0 0 1\
|0 1 0|  |0 1 0|  |0 1 0|
\0 0 1/  \0 0 1/  \0 0 1/
/1 0 0\  /0 1 0\  /0 0 1\
|0 0 1|  |0 0 1|  |0 0 1|
\0 0 1/  \0 0 1/  \0 0 1/

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, p. 42.

Seiichi Manyama, Table of n, a(n) for n = 0..200
Hsien-Kuei Hwang, Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019.

Cf. A158691, A179525, A289312, A289313, A289314, A289315, A289317.

C:= x -> x/(1 - x^2):
G:= add(mul( (1 + C(x))^k - 1, k=1..n),n=0..20):
S:= series(G,x,21):
seq(coeff(S,x,j),j=0..20);

G.f.: A(x) = Sum_{n >= 0} Product_{k = 1..n} ( (1 + x/(1 - x^2))^k - 1 ).

a(n) ~ 12^(n+1) * n^(n + 1/2) / (exp(n + Pi^2/24) * Pi^(2*n + 3/2)). - Vaclav Kotesovec, Aug 31 2023

A182319 Triangular array: T(n,k) counts upper triangular matrices with entries from {0,1} having n 1's in total, with k 1's on the main diagonal and at least one nonzero entry in each row.

Original entry on oeis.org

1, 1, 1, 2, 4, 1, 7, 16, 9, 1, 33, 83, 64, 16, 1, 197, 530, 486, 180, 25, 1, 1419, 4026, 4144, 1930, 410, 36, 1, 11966, 35462, 39746, 21574, 5965, 812, 49, 1, 115575, 355368, 425762, 258426, 85589, 15477, 1456, 64, 1

Offset: 1

Peter Bala, Apr 24 2012

This triangle is a refinement of A179525.

			Triangle begins
.n\k.|...1.....2.....3.....4.....5
= = = = = = = = = = = = = = = = = =
..1..|...1
..2..|...1.....1
..3..|...2.....4.....1
..4..|...7....16.....9.....1
..5..|..33....83....64....16.....1
..6..|.197...530...486...180....25.....1
...
T(3,2) = 4: there is one 2x2 matrix and three 3x3 matrices with the specified properties:
........1..0..0.....0..1..0.....0..0..1..
1.1.....0..0..1.....0..1..0.....0..1..0..
0.1.....0..0..1.....0..0..1.....0..0..1..

Andrew Howroyd, Table of n, a(n) for n = 1..1275

A179525.

\\ A(n) returns vector of n'th row.
A(n)={ my(rv=if(n==1, [1], vector(n)));
my(M=matrix(2,n,k,s,k==2&&s==1));
\\ M[k,s] is number of configs with s 1's with k+1 on diagonal.
for(r=2, n, M=matrix(r+1,n,k,s, sum(j=0, min(s-1,r-1), binomial(r-1,j) * (if(j>0&&k<=r, M[k,s-j]) + if(j1, M[k-1,s-j-1]))));
  for(i=1, r, rv[i]+=M[i+1,n])); rv }
for(n=1,10,print(A(n))); \\ Andrew Howroyd, Oct 10 2017

Let F(x,t) = 1 - (1-t)*(1 - (1-x*t) + (1-x*t)*(1-x^2*t) - (1-x*t)*(1-x^2*t)*(1-x^3*t) + ...). Then F(1+x,1+x*t) = 1 + x*t + (t+t^2)*x^2 + (2*t+4*t^2+t^3)*x^3 + (7*t+16*t^2+9*t^3+t^4)*x^4 + ... is conjecturally a generating function for the triangle.
T(n+1,1) = sum {k = 1..n} T(n,k); T(n+1,n) = n^2.
First column A179525. Row sums A179525.

Terms a(23) and beyond from Andrew Howroyd, Oct 10 2017

cp's OEIS Frontend

A186737 G.f. satisfies: A(x) = Sum_{n>=0} Product_{k=1..n} ((1 + x*A(x))^k - 1).

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A207434 L.g.f.: log( Sum_{n>=0} Product_{k=1..n} ((1+x)^k - 1) ) = Sum_{n>=1} a(n)*x^n/n.

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A207649 E.g.f.: Sum_{n>=0} 1/n! * Product_{k=1..n} ((1+x)^k - 1).

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A289316 The number of upper-triangular matrices whose nonzero entries are positive odd numbers summing to n and each row contains a nonzero entry.

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A182319 Triangular array: T(n,k) counts upper triangular matrices with entries from {0,1} having n 1's in total, with k 1's on the main diagonal and at least one nonzero entry in each row.

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