A204064 -id:A204064 - cp's OEIS frontend

A229046 G.f.: Sum_{n>=0} n! * x^n * (1+x)^n / Product_{k=1..n} (1 + k*x).

1, 1, 2, 4, 10, 28, 88, 304, 1144, 4648, 20248, 94024, 463144, 2409928, 13198888, 75848584, 456066664, 2862257608, 18708144808, 127096142344, 895846801384, 6540722530888, 49392459602728, 385251753351304, 3099780861286504, 25698921466247368, 219294936264513448

Offset: 0

Paul D. Hanna, Oct 27 2013

nonn

a(n-1) is the number of set partitions of [n] such that the absolute difference between least elements of consecutive blocks is always > 1. a(4) = 10: 12345, 1234|5, 1235|4, 123|45, 1245|3, 124|35, 124|3|5, 125|34, 12|345, 12|34|5. - Alois P. Heinz, May 22 2017

Conjecturally, the number of sequences (e(1), ..., e(n)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i) = e(k). [Martinez and Savage, 2.13] - Eric M. Schmidt, Jul 17 2017

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 28*x^5 + 88*x^6 + 304*x^7 +...
where
A(x) = 1 + x*(1+x)/(1+x) + 2!*x^2*(1+x)^2/((1+x)*(1+2*x)) + 3!*x^3*(1+x)^3/((1+x)*(1+2*x)*(1+3*x)) + 4!*x^4*(1+x)^4/((1+x)*(1+2*x)*(1+3*x)*(1+4*x)) + 5!*x^5*(1+x)^5/((1+x)*(1+2*x)*(1+3*x)*(1+4*x)*(1+5*x)) +...
Also, we have the identity (cf. A204064):
A(x) = 1 + x + 2*x^2*(1+x)/(1+x+x^2) + 2*x^3*(1+2*x)*(2+2*x)/((1+x+2*x^2)*(1+2*x+2*x^2)) + 2*x^4*(1+3*x)*(2+3*x)*(3+3*x)/((1+x+3*x^2)*(1+2*x+3*x^2)*(1+3*x+3*x^2)) + 2*x^5*(1+4*x)*(2+4*x)*(3+4*x)*(4+4*x)/((1+x+4*x^2)*(1+2*x+4*x^2)*(1+3*x+4*x^2)*(1+4*x+4*x^2)) +...
Also, by Peter Bala's o.g.f.:
A(x) = 1/((1+x)*(1-x)) + x/((1+x)^2*(1-2*x)) + x^2/((1+x)^3*(1-3*x))+ x^3/((1+x)^4*(1-4*x))+ x^4/((1+x)^5*(1-5*x)) + x^5/((1+x)^6*(1-6*x)) +...

Paul D. Hanna, Table of n, a(n) for n = 0..500
Wenqin Cao, Emma Yu Jin, and Zhicong Lin, Enumeration of inversion sequences avoiding triples of relations, Discrete Applied Mathematics (2019); see also author's copy
Megan A. Martinez and Carla D. Savage, Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations, arXiv:1609.08106 [math.CO], 2016.

Cf. A204064, A105795, A112532, A245373, A298668.

a:= n-> add(k!*Stirling2(n-k+1,k+1), k=0..floor(n/2)):
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 24 2018

a[n_] := Sum[k!*StirlingS2[n-k+1, k+1], {k, 0, n/2}];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jul 25 2018, after Alois P. Heinz *)

a(n)=polcoeff( sum(m=0, n, m!*x^m*(1+x)^m/prod(k=1, m, 1+k*x +x*O(x^n))), n)
for(n=0,30,print1(a(n),", "))

a(n)=polcoeff( 1-x + 2*x*sum(m=0, n, x^m*prod(k=1, m, (k+m*x)/(1+k*x+m*x^2 +x*O(x^n))) ), n)
for(n=0,30,print1(a(n), ", "))

/* After Peter Bala: Sum_{n>=0} x^n/((1+x)^(n+1)*(1 - (n+1)*x)) */
{a(n)=polcoeff( sum(m=0, n, x^m/((1+x)^(m+1)*(1 - (m+1)*x) +x*O(x^n))), n)} \\ Paul D. Hanna, Jul 13 2014
for(n=0,30,print1(a(n), ", "))

a(n)=sum(k=0, floor(n/2), sum(i=0, k, (-1)^i*binomial(k,i)*(k-i+1)^(n-k)))
for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Jul 13 2014

G.f.: 1+x + Sum_{n>=1} 2*x^(n+1) * Product_{k=1..n} (k + n*x)/(1 + k*x + n*x^2).
From Peter Bala, Jul 09 2014: (Start)
An alternative form of the o.g.f. appears to be the formal series A(x) = 1/(1 + x) * Sum_{n >= 0} 1/(1 - (n+1)*x)*(x/(1 + x))^n (checked up to a(26)). Cf. A105795.
Setting y = x/(1 + x) produces A(y) = (1 - y)^2*( Sum_{n >= 0} y^n/(1 - (n + 2)*y) ) = 1 + y + 3*y^2 + 9*y^3 + ..., the generating function for A112532. (End)
a(n) = 2*A204064(n-1) for n>1.
a(n) = Sum_{k=0..floor(n/2)} Sum_{i=0..k} (-1)^i * binomial(k,i) * (k-i+1)^(n-k). (See Paul Barry's formula in A105795). - Paul D. Hanna, Jul 13 2014
From Alois P. Heinz, Jan 24 2018: (Start)
a(n) = Sum_{k=0..floor(n/2)} k! * Stirling2(n-k+1,k+1).
a(n) = Sum_{k=1..ceiling((n+1)/2)} A298668(n+1,k). (End)

A187741 G.f.: Sum_{n>=0} (1 + nx)^n x^n / (1 + x + n*x^2)^n.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 12, 24, 60, 120, 360, 720, 2520, 5040, 20160, 40320, 181440, 362880, 1814400, 3628800, 19958400, 39916800, 239500800, 479001600, 3113510400, 6227020800, 43589145600, 87178291200, 653837184000, 1307674368000, 10461394944000, 20922789888000

Offset: 0

Paul D. Hanna, Jan 03 2013

nonn

This is an enumeration of the disjoint union (with repetition) of A001710(n), for n > 0, and A000142(n), for n > 0. The first lists the orders of the alternating groups; the second lists the orders of the symmetric groups. - Hal M. Switkay, Mar 13 2023

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 6*x^5 + 12*x^6 + 24*x^7 + 60*x^8 +...
where
A(x) = 1 + (1+x)*x/(1+x+x^2) + (1+2*x)^2*x^2/(1+x+2*x^2)^2 + (1+3*x)^3*x^3/(1+x+3*x^2)^3 + (1+4*x)^4*x^4/(1+x+4*x^2)^4 + (1+5*x)^5*x^5/(1+x+5*x^2)^5 +...

Cf. A000142, A001710, A187742, A187735, A208236, A204064, A361382.

a[n_] := If[OddQ[n], ((n + 1)/2)!, (n/2 + 1)!/2]; a[0] = 1; Array[a, 32, 0] (* Amiram Eldar, Dec 11 2022 *)

{a(n)=polcoeff( sum(m=0, n, (x+m*x^2)^m / (1 + x+m*x^2 +x*O(x^n))^m), n)}
for(n=0, 40, print1(a(n), ", "))

{a(n)=if(n==0,1, if(n%2==0, ((n+2)/2)!/2, ((n+1)/2)! ))}
for(n=0, 30, print1(a(n), ", "))

G.f.: 1/2 + (1+2*x) * Sum_{n>=0} (n+1)!*x^(2*n)/2.
a(2*n) = (n+1)!/2,  a(2*n-1) = n!,  for n>0 with a(0)=1.
From Amiram Eldar, Dec 11 2022: (Start)
Sum_{n>=0} 1/a(n) = 3*e - 4.
Sum_{n>=0} (-1)^n/a(n) = e - 2. (End)

A208237 G.f.: Sum_{n>=0} n! * x^n * Product_{k=1..n} (1 + kx) / (1 + kx + k^2*x^2).

Original entry on oeis.org

1, 1, 2, 5, 15, 54, 223, 1045, 5474, 31685, 200895, 1384470, 10304431, 82376101, 703949762, 6403761365, 61784985615, 630180031734, 6775001385343, 76572619018165, 907658144193314, 11259399965148005, 145879271404693215, 1970471655222795990, 27702625497930064591

Offset: 0

Paul D. Hanna, Jan 11 2013

nonn

Compare to the identity:
Sum_{n>=0} n! * x^n * Product_{k=1..n} (1 + x) / (1 + k*x + k*x^2) = 1/(1-x-x^2).
Compare also to the g.f. of A136127:
x*Sum_{n>=0} n! * x^n * Product_{k=1..n} (2 + k*x) / (1 + 2*k*x + k^2*x^2).

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + 54*x^5 + 223*x^6 + 1045*x^7 +...
where
A(x) = 1 + x*(1+x)/(1+x+x^2) + 2!*x^2*(1+x)*(1+2*x)/((1+x+x^2)*(1+2*x+4*x^2)) + 3!*x^3*(1+x)*(1+2*x)*(1+3*x)/((1+x+x^2)*(1+2*x+4*x^2)*(1+3*x+9*x^2)) + 4!*x^4*(1+x)*(1+2*x)*(1+3*x)*(1+4*x)/((1+x+x^2)*(1+2*x+4*x^2)*(1+3*x+9*x^2)*(1+4*x+16*x^2)) +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..270

Cf. A221370, A204064, A204066, A208236, A136127.

{a(n)=polcoeff( sum(m=0, n, m!*x^m*prod(k=1, m, (1+k*x)/(1+k*x+k^2*x^2 +x*O(x^n))) ), n)}
for(n=0, 30, print1(a(n), ", "))

a(n) ~ 2 * 3^(n/2 + 5/4) * n^(n+2) / (exp(n) * Pi^(n+3/2)). - Vaclav Kotesovec, Nov 02 2014

A204066 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (n + kx) / (1 + nx + k*x^2).

Original entry on oeis.org

1, 1, 4, 16, 82, 502, 3574, 29002, 264166, 2668666, 29612014, 358025986, 4684916902, 65966957722, 994546450174, 15984888286642, 272845934899606, 4929166716321706, 93963635086523374, 1884915966747571906, 39691711412770983622, 875410001054417122042, 20180907494704416823774

Offset: 0

Paul D. Hanna, Jan 09 2013

nonn

			G.f.: A(x) = 1 + x + 4*x^2 + 16*x^3 + 82*x^4 + 502*x^5 + 3574*x^6 +...
where
A(x) = 1 + x*(1+x)/(1+x+x^2) + x^2*(2+x)*(2+2*x)/((1+2*x+x^2)*(1+2*x+2*x^2)) + x^3*(3+x)*(3+2*x)*(3+3*x)/((1+3*x+x^2)*(1+3*x+2*x^2)*(1+3*x+3*x^2)) + x^4*(4+x)*(4+2*x)*(4+3*x)*(4+4*x)/((1+4*x+x^2)*(1+4*x+2*x^2)*(1+4*x+3*x^2)*(1+4*x+4*x^2)) +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..250

Cf. A204064, A187741, A187742, A220181.

{a(n)=polcoeff( sum(m=0, n, x^m*prod(k=1, m, (m+k*x)/(1+m*x+k*x^2 +x*O(x^n))) ), n)}
for(n=0, 30, print1(a(n), ", "))

a(n) ~ exp(1/2) * n! * n/2. - Vaclav Kotesovec, Nov 02 2014

A208236 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (1 + nkx) / (1 + x + nkx^2).

Original entry on oeis.org

1, 1, 1, 4, 10, 50, 208, 1290, 7456, 55982, 411796, 3650514, 32484460, 332970374, 3468625588, 40420787250, 481757564956, 6295577910182, 84407459209876, 1223095585594674, 18208380720893980, 289843786627539014, 4741844351895315028, 82269590167564595250

Offset: 0

Paul D. Hanna, Jan 10 2013

nonn

Compare to the identity:
Sum_{n>=0} x^n * Product_{k=1..n} (1 + t*k*x) / (1 + x + t*k*x^2) = (1+x)/(1-t*x^2).
Compare to the g.f. of A187741:
Sum_{n>=0} x^n * (1 + n*x)^n / (1 + x + n*x^2)^n  =  1/2 + (1+2*x)*Sum_{n>=0} (n+1)!*x^(2*n)/2.
Limit n->infinity (a(n)/n!)^(1/n) = 1/(2*log(2)). - Vaclav Kotesovec, Nov 03 2014

			G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 10*x^4 + 50*x^5 + 208*x^6 + 1290*x^7 +...
where
A(x) = 1 + x*(1+x)/(1+x+x^2) + x^2*(1+2*1*x)*(1+2*2*x)/((1+x+2*1*x^2)*(1+x+2*2*x^2)) + x^3*(1+3*1*x)*(1+3*2*x)*(1+3*3*x)/((1+x+3*1*x^2)*(1+x+3*2*x^2)*(1+x+3*3*x^2)) + x^4*(1+4*1*x)*(1+4*2*x)*(1+4*3*x)*(1+4*4*x)/((1+x+4*1*x^2)*(1+x+4*2*x^2)*(1+x+4*3*x^2)*(1+x+4*4*x^2)) +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Cf. A204064, A187741, A204066.

{a(n)=polcoeff( sum(m=0, n, x^m*prod(k=1, m, (1+m*k*x)/(1+x+m*k*x^2 +x*O(x^n))) ), n)}
for(n=0, 30, print1(a(n), ", "))

A231291 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (1 + kx) / (1 - x - kx^2).

Original entry on oeis.org

1, 1, 3, 9, 29, 99, 355, 1333, 5213, 21163, 88899, 385413, 1720637, 7894827, 37166563, 179254501, 884548253, 4460597131, 22962705027, 120557527941, 644952640253, 3512995320939, 19468234666531, 109694091843109, 628027149163613, 3651429293510731, 21547912967252163

Offset: 0

Paul D. Hanna, Nov 06 2013

nonn

Compare to the identity:

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

			G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 29*x^4 + 99*x^5 + 355*x^6 + 1333*x^7 +...
where
A(x) = 1 + x*(1+x)/(1-x-x^2) + x^2*(1+x)*(1+2*x)/((1-x-x^2)*(1-x-2*x^2)) + x^3*(1+x)*(1+2*x)*(1+3*x)/((1-x-x^2)*(1-x-2*x^2)*(1-x-3*x^2)) + x^4*(1+x)*(1+2*x)*(1+3*x)*(1+4*x)/((1-x-x^2)*(1-x-2*x^2)*(1-x-3*x^2)*(1-x-4*x^2)) +...

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

Cf. A204064, A231274.

{a(n)=polcoeff( sum(m=0, n, x^m*prod(k=1, m, (1+k*x)/(1-x-k*x^2 +x*O(x^n))) ), n)}
for(n=0, 30, print1(a(n), ", "))

A227851 G.f.: Sum_{n>=1} n^n * x^n * Product_{k=1..n} (k + x) / (1 + nkx + n*x^2).

Original entry on oeis.org

1, 1, 8, 125, 3650, 171440, 11815940, 1122759980, 140645621840, 22456283261240, 4451225265169640, 1072410309912462440, 308628265617560695880, 104567048162852196877640, 41198829781936190483346440, 18676924223093561435394148040, 9652952812685808726911849225480

Offset: 0

Paul D. Hanna, Oct 31 2013

nonn

Compare to the identity:

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

			G.f.: A(x) = 1 + x + 8*x^2 + 125*x^3 + 3650*x^4 + 171440*x^5 +...
where
A(x) = 1 + x*(1+x)/(1+x+x^2) + 2^2*x^2*(1+x)*(2+x)/((1+2*x+2*x^2)*(1+4*x+2*x^2)) + 3^3*x^3*(1+x)*(2+x)*(3+x)/((1+3*x+3*x^2)*(1+6*x+3*x^2)*(1+9*x+3*x^2)) + 4^4*x^4*(1+x)*(2+x)*(3+x)*(4+x)/((1+4*x+4*x^2)*(1+8*x+4*x^2)*(1+12*x+4*x^2)*(1+16*x+4*x^2)) +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..236

Cf. A204064.

{a(n)=polcoeff( sum(m=0, n, m^m*x^m*prod(k=1, m, (k+x)/(1+m*k*x+m*x^2 +x*O(x^n))) ), n)}
for(n=0, 20, print1(a(n), ", "))

a(n) == 2 (mod 3) for n>1.
a(n) == 0 (mod 5) for n>2.
a(n) == 0 (mod 8) for n>7.
a(n) ~ sqrt(Pi) * n^(2*n+1/2) / (sqrt(1-log(2)) * exp(2*n) * (log(2))^(2*n+1)). - Vaclav Kotesovec, Nov 03 2014

A363110 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (k + (n-k+1)x) / (1 + kx + (n-k+1)*x^2).

Original entry on oeis.org

1, 1, 2, 4, 10, 28, 88, 306, 1158, 4730, 20722, 96776, 479340, 2507510, 13804014, 79718782, 481614806, 3036358968, 19932689952, 135981543762, 962319171782, 7053068549250, 53458038451082, 418440466421960, 3378290373259300, 28099682071640734, 240537280709926718

Offset: 0

Paul D. Hanna, Jun 02 2023

nonn

Compare to the following identities, which hold for any fixed b and c:
(1) Sum_{n>=0} x^n * Product_{k=1..n} (b + k*x)/(1 + b*x + k*x^2) = (1 + b*x)/(1 - x^2).
(2) Sum_{n>=0} x^n * Product_{k=1..n} (k + c*x)/(1 + k*x + c*x^2) = (1 + c*x^2)/(1 - x).
(3) Sum_{n>=0} x^n * Product_{k=1..n} (b*k + c*k*x)/(1 + b*k*x + c*k*x^2) = 1/(1 - b*x - c*x^2).
Conjectures:
(1) a(6*n + k) == 0 (mod 4) for n > 0 when k = {0,5},
(2) a(6*n + k) == 2 (mod 4) for n > 0 when k = {1,2,3,4}.

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 28*x^5 + 88*x^6 + 306*x^7 + 1158*x^8 + 4730*x^9 + 20722*x^10 + 96776*x^11 + 479340*x^12 + ...
where
A(x) = 1 + x*(1+x)/(1+x+x^2) + x^2*(1 + 2*x)*(2 + x)/((1 + x + 2*x^2)*(1 + 2*x + x^2)) + x^3*(1 + 3*x)*(2 + 2*x)*(3 + x)/((1 + x + 3*x^2)*(1 + 2*x + 2*x^2)*(1 + 3*x + x^2)) + x^4*(1 + 4*x)*(2 + 3*x)*(3 + 2*x)*(4 + x)/((1 + x + 4*x^2)*(1 + 2*x + 3*x^2)*(1 + 3*x + 2*x^2)*(1 + 4*x + x^2)) + ...

Paul D. Hanna, Table of n, a(n) for n = 0..400

Cf. A204064, A204066, A316370, A067948.

{a(n) = polcoeff( A = sum(m=0, n, x^m*prod(k=1, m, (k + (m-k+1)*x)/(1 + k*x + (m-k+1)*x^2 +x*O(x^n))) ), n)}
for(n=0, 30, print1(a(n), ", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^n may be described by the following.
(1) Sum_{n>=0} x^n * Product_{k=1..n} (k + (n-k+1)*x) / (1 + k*x + (n-k+1)*x^2).
(2) Sum_{n>=0} x^n * (Sum_{k=0..n} A067948(n,k) * x^k) / Product_{k=1..n} (1 + k*x + (n-k+1)*x^2).

cp's OEIS Frontend

A229046 G.f.: Sum_{n>=0} n! * x^n * (1+x)^n / Product_{k=1..n} (1 + k*x).

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A187741 G.f.: Sum_{n>=0} (1 + n*x)^n * x^n / (1 + x + n*x^2)^n.

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A208237 G.f.: Sum_{n>=0} n! * x^n * Product_{k=1..n} (1 + k*x) / (1 + k*x + k^2*x^2).

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A204066 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (n + k*x) / (1 + n*x + k*x^2).

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A208236 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (1 + n*k*x) / (1 + x + n*k*x^2).

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A231291 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (1 + k*x) / (1 - x - k*x^2).

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A227851 G.f.: Sum_{n>=1} n^n * x^n * Product_{k=1..n} (k + x) / (1 + n*k*x + n*x^2).

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A187741 G.f.: Sum_{n>=0} (1 + nx)^n x^n / (1 + x + n*x^2)^n.

A208237 G.f.: Sum_{n>=0} n! * x^n * Product_{k=1..n} (1 + kx) / (1 + kx + k^2*x^2).

A204066 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (n + kx) / (1 + nx + k*x^2).

A208236 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (1 + nkx) / (1 + x + nkx^2).

A231291 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (1 + kx) / (1 - x - kx^2).

A227851 G.f.: Sum_{n>=1} n^n * x^n * Product_{k=1..n} (k + x) / (1 + nkx + n*x^2).

A363110 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (k + (n-k+1)x) / (1 + kx + (n-k+1)*x^2).