A245373 -id:A245373 - 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)

A245374 G.f.: Sum_{n>=0} x^n / ( (1+x)^(n+1) * (1 - 3(n+1)x) ).

Original entry on oeis.org

1, 3, 12, 54, 288, 1782, 12474, 96714, 819882, 7536402, 74610234, 790692354, 8921660922, 106687646802, 1346863560714, 17890362862434, 249297686894682, 3634756665823602, 55317506662094634, 876911386062810114, 14451743847813157242, 247171758180997987602, 4380263376360686471754

Offset: 0

Paul D. Hanna, Jul 19 2014

nonn

			G.f.: A(x) = 1 + 3*x + 12*x^2 + 54*x^3 + 288*x^4 + 1782*x^5 + 12474*x^6 +...
where we have the following series identity:
A(x) = 1/((1+x)*(1-3*x)) + x/((1+x)^2*(1-6*x)) + x^2/((1+x)^3*(1-9*x))+ x^3/((1+x)^4*(1-12*x))+ x^4/((1+x)^5*(1-15*x)) + x^5/((1+x)^6*(1-18*x)) +...
is equal to
A(x) = 1 + 3*x*(1+x)/(1+3*x) + 2!*(3*x)^2*(1+x)^2/((1+3*x)*(1+6*x)) + 3!*(3*x)^3*(1+x)^3/((1+3*x)*(1+6*x)*(1+9*x)) + 4!*(3*x)^4*(1+x)^4/((1+3*x)*(1+6*x)*(1+9*x)*(1+12*x)) + 5!*(3*x)^5*(1+x)^5/((1+3*x)*(1+6*x)*(1+9*x)*(1+12*x)*(1+15*x)) +...

Cf. A229046, A245373, A245375, A245376.

{a(n)=polcoeff( sum(m=0, n, x^m/((1+x)^(m+1)*(1 - 3*(m+1)*x) +x*O(x^n))), n)}
for(n=0, 30, print1(a(n), ", "))

{a(n)=polcoeff( sum(m=0, n, 3^m*m!*x^m*(1+x)^m/prod(k=1, m, 1+3*k*x +x*O(x^n))), n)}
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)*3^(n-k)))}
for(n=0, 30, print1(a(n), ", "))

G.f.: Sum_{n>=0} n! * (3*x)^n * (1+x)^n / Product_{k=1..n} (1 + 3*k*x).

a(n) = Sum_{k=0..floor(n/2)} Sum_{i=0..k} (-1)^i * binomial(k,i) * 3^(n-k) * (k-i+1)^(n-k).

A245375 G.f.: Sum_{n>=0} x^n / ( (1+x)^(n+1) * (1 - 4(n+1)x) ).

Original entry on oeis.org

1, 4, 20, 112, 736, 5632, 49024, 474112, 5017600, 57597952, 712597504, 9446981632, 133474877440, 2000265674752, 31666683510784, 527786775150592, 9233419259084800, 169106747636580352, 3234542505882025984, 64473076850860490752, 1336621867385969704960, 28769619371258703511552

Offset: 0

Paul D. Hanna, Jul 19 2014

nonn

			G.f.: A(x) = 1 + 4*x + 20*x^2 + 112*x^3 + 736*x^4 + 5632*x^5 + 49024*x^6 +...
where we have the following series identity:
A(x) = 1/((1+x)*(1-4*x)) + x/((1+x)^2*(1-8*x)) + x^2/((1+x)^3*(1-12*x))+ x^3/((1+x)^4*(1-16*x))+ x^4/((1+x)^5*(1-20*x)) + x^5/((1+x)^6*(1-24*x)) +...
is equal to
A(x) = 1 + 4*x*(1+x)/(1+4*x) + 2!*(4*x)^2*(1+x)^2/((1+4*x)*(1+8*x)) + 3!*(4*x)^3*(1+x)^3/((1+4*x)*(1+8*x)*(1+12*x)) + 4!*(4*x)^4*(1+x)^4/((1+4*x)*(1+8*x)*(1+12*x)*(1+16*x)) + 5!*(4*x)^5*(1+x)^5/((1+4*x)*(1+8*x)*(1+12*x)*(1+16*x)*(1+20*x)) +...

Cf. A229046, A245373, A245374, A245376.

{a(n)=polcoeff( sum(m=0, n, x^m/((1+x)^(m+1)*(1 - 4*(m+1)*x) +x*O(x^n))), n)}
for(n=0, 30, print1(a(n), ", "))

{a(n)=polcoeff( sum(m=0, n, 4^m*m!*x^m*(1+x)^m/prod(k=1, m, 1+4*k*x +x*O(x^n))), n)}
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)*4^(n-k)))}
for(n=0, 30, print1(a(n), ", "))

G.f.: Sum_{n>=0} n! * (4*x)^n * (1+x)^n / Product_{k=1..n} (1 + 4*k*x).

a(n) = Sum_{k=0..floor(n/2)} Sum_{i=0..k} (-1)^i * binomial(k,i) * 4^(n-k) * (k-i+1)^(n-k).

A245376 G.f.: Sum_{n>=0} x^n / ( (1+x)^(n+1) * (1 - 5(n+1)x) ).

Original entry on oeis.org

1, 5, 30, 200, 1550, 14000, 144500, 1662500, 20952500, 286437500, 4221312500, 66703437500, 1124194062500, 20109785937500, 380209901562500, 7571141773437500, 158312671414062500, 3466819503710937500, 79316483272226562500, 1891747084452148437500, 46942864023040039062500

Offset: 0

Paul D. Hanna, Jul 19 2014

nonn

			G.f.: A(x) = 1 + 5*x + 30*x^2 + 200*x^3 + 1550*x^4 + 14000*x^5 +...
where we have the following series identity:
A(x) = 1/((1+x)*(1-5*x)) + x/((1+x)^2*(1-10*x)) + x^2/((1+x)^3*(1-15*x))+ x^3/((1+x)^4*(1-20*x))+ x^4/((1+x)^5*(1-25*x)) + x^5/((1+x)^6*(1-30*x)) +...
is equal to
A(x) = 1 + 5*x*(1+x)/(1+5*x) + 2!*(5*x)^2*(1+x)^2/((1+5*x)*(1+10*x)) + 3!*(5*x)^3*(1+x)^3/((1+5*x)*(1+10*x)*(1+15*x)) + 4!*(5*x)^4*(1+x)^4/((1+5*x)*(1+10*x)*(1+15*x)*(1+20*x)) + 5!*(5*x)^5*(1+x)^5/((1+5*x)*(1+10*x)*(1+15*x)*(1+20*x)*(1+25*x)) +...

Cf. A229046, A245373, A245374, A245375.

{a(n)=polcoeff( sum(m=0, n, x^m/((1+x)^(m+1)*(1 - 5*(m+1)*x) +x*O(x^n))), n)}
for(n=0, 30, print1(a(n), ", "))

{a(n)=polcoeff( sum(m=0, n, 5^m*m!*x^m*(1+x)^m/prod(k=1, m, 1+5*k*x +x*O(x^n))), n)}
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)*5^(n-k)))}
for(n=0, 30, print1(a(n), ", "))

G.f.: Sum_{n>=0} n! * (5*x)^n * (1+x)^n / Product_{k=1..n} (1 + 5*k*x).

a(n) = Sum_{k=0..floor(n/2)} Sum_{i=0..k} (-1)^i * binomial(k,i) * 5^(n-k) * (k-i+1)^(n-k).

A245378 G.f. satisfies: A(x) = Sum_{n>=0} x^n / ( (1+x)^(n+1) * (1 - (n+1)xA(x)) ).

Original entry on oeis.org

1, 1, 3, 10, 39, 165, 743, 3507, 17199, 87126, 454159, 2430031, 13326623, 74856230, 430628069, 2538270783, 15343363603, 95233568052, 607850790015, 3996223189468, 27105153736781, 189947851239185, 1376864409041417, 10330672337146804, 80248762443834399, 645206035074873681

Offset: 0

Paul D. Hanna, Jul 19 2014

nonn

Compare g.f. to an identity for C(x) = 1 + x*C(x)^2, the Catalan function:

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

			G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 39*x^4 + 165*x^5 + 743*x^6 +...
where we have the following identity:
A(x) = 1/((1+x)*(1-x*A(x))) + x/((1+x)^2*(1-2*x*A(x))) + x^2/((1+x)^3*(1-3*x*A(x)))+ x^3/((1+x)^4*(1-4*x*A(x)))+ x^4/((1+x)^5*(1-5*x*A(x))) + x^5/((1+x)^6*(1-6*x*A(x))) +...
is equal to
A(x) = 1 + x*A(x)*(1+x)/(1+x*A(x)) + 2!*x^2*A(x)^2*(1+x)^2/((1+x*A(x))*(1+2*x*A(x))) + 3!*x^3*A(x)^3*(1+x)^3/((1+x*A(x))*(1+2*x*A(x))*(1+3*x*A(x))) + 4!*x^4*A(x)^4*(1+x)^4/((1+x*A(x))*(1+2*x*A(x))*(1+3*x*A(x))*(1+4*x*A(x))) + 5!*x^5*A(x)^5*(1+x)^5/((1+x*A(x))*(1+2*x*A(x))*(1+3*x*A(x))*(1+4*x*A(x))*(1+5*x*A(x))) +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..175

Cf. A229046, A245373, A245374, A245375, A245376.

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

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

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

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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A245374 G.f.: Sum_{n>=0} x^n / ( (1+x)^(n+1) * (1 - 3*(n+1)*x) ).

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A245375 G.f.: Sum_{n>=0} x^n / ( (1+x)^(n+1) * (1 - 4*(n+1)*x) ).

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A245376 G.f.: Sum_{n>=0} x^n / ( (1+x)^(n+1) * (1 - 5*(n+1)*x) ).

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A245378 G.f. satisfies: A(x) = Sum_{n>=0} x^n / ( (1+x)^(n+1) * (1 - (n+1)*x*A(x)) ).

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A245374 G.f.: Sum_{n>=0} x^n / ( (1+x)^(n+1) * (1 - 3(n+1)x) ).

A245375 G.f.: Sum_{n>=0} x^n / ( (1+x)^(n+1) * (1 - 4(n+1)x) ).

A245376 G.f.: Sum_{n>=0} x^n / ( (1+x)^(n+1) * (1 - 5(n+1)x) ).

A245378 G.f. satisfies: A(x) = Sum_{n>=0} x^n / ( (1+x)^(n+1) * (1 - (n+1)xA(x)) ).