A121689 -id:A121689 - cp's OEIS frontend

A121690 G.f.: A(x) = Sum_{k>=0} x^k * (1+x)^(k*(k+1)/2).

1, 1, 2, 4, 10, 27, 81, 262, 910, 3363, 13150, 54135, 233671, 1053911, 4951997, 24177536, 122381035, 640937746, 3466900453, 19337255086, 111057640382, 655892813805, 3978591077096, 24760700544301, 157941950878839

Offset: 0

Paul D. Hanna, Aug 15 2006

nonn

a(n) is the number of length n permutations that simultaneously avoid the bivincular patterns (123,{2},{}) and (132,{},{2}). - Christian Bean, Jun 03 2015

Seiichi Manyama, Table of n, a(n) for n = 0..685
Christian Bean, A Claesson, H Ulfarsson, Simultaneous Avoidance of a Vincular and a Covincular Pattern of Length 3, arXiv preprint arXiv:1512.03226, 2015

Cf. A121689, A131338.

Table[Sum[Binomial[k*(k+1)/2,n-k],{k,0,n}],{n,0,30}] (* Vaclav Kotesovec, Jun 03 2015 *)

a(n) = sum(k=0,n, binomial(k*(k+1)/2, n-k))
for(n=0, 30, print1(a(n), ", "))

{a(n)=polcoeff(sum(m=0, n, x^m*prod(k=1, m, (1 - x*(1+x)^(2*k-2))/(1 - x*(1+x)^(2*k-1) + x*O(x^n)))), n)}
for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Mar 21 2018

a(n) = Sum_{k=0..n} C(k*(k+1)/2,n-k).
a(n) = A131338(n+1, n*(n+1)/2 + 1) for n>=0, where triangle A131338 starts with a '1' in row 0 and then for n>0 row n consists of n '1's followed by the partial sums of the prior row. - Paul D. Hanna, Aug 30 2007
From Paul D. Hanna, Apr 24 2010: (Start)
Let q = (1+x), then g.f. A(x) equals the continued fraction:
A(x) = 1/(1 - q*x/(1 - (q^2-q)*x/(1 - q^3*x/(1 - (q^4-q^2)*x/(1 - q^5*x/(1- (q^6-q^3)*x/(1 - q^7*x/(1 - (q^8-q^4)*x/(1 - ...)))))))))
due to an identity of a partial elliptic theta function.
(End)
G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (1 - x*(1+x)^(2*k-2))/(1 - x*(1+x)^(2*k-1)). - Paul D. Hanna, Mar 21 2018
log(a(n)) ~ n*log(n) - 2*n*log(log(n)) - n*(1 - log(2)) + 4*n*log(log(n))/log(n) - 2*n*log(2)/log(n). - Vaclav Kotesovec, Jul 01 2025

A325580 G.f.: A(x,y) = Sum_{n>=0} x^n * ((1+x)^n + y)^n, where A(0) = 0, as a triangle of coefficients T(n,k) of x^ny^k in A(x,y) = Sum_{n>=0} x^n Sum_{k=0..n} T(n,k)x^ny^k, read by rows.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 5, 7, 3, 1, 16, 24, 15, 4, 1, 57, 98, 67, 26, 5, 1, 231, 430, 336, 144, 40, 6, 1, 1023, 2062, 1767, 861, 265, 57, 7, 1, 4926, 10610, 9873, 5300, 1845, 440, 77, 8, 1, 25483, 58240, 58221, 33974, 13041, 3501, 679, 100, 9, 1, 140601, 338984, 360930, 226716, 94580, 27978, 6083, 992, 126, 10, 1, 822422, 2081189, 2345469, 1572134, 706225, 226843, 54271, 9886, 1389, 155, 11, 1, 5074015, 13423258, 15926115, 11318196, 5428820, 1876728, 486941, 97448, 15246, 1880, 187, 12, 1

Offset: 0

Paul D. Hanna, May 11 2019

			G.f. A(x,y) = Sum_{n>=0} x^n * Sum_{k=0..n} T(n,k)*x^n*y^k begins:
A(x,y) = 1 + (y + 1)*x + (y^2 + 2*y + 2)*x^2 + (y^3 + 3*y^2 + 7*y + 5)*x^3 + (y^4 + 4*y^3 + 15*y^2 + 24*y + 16)*x^4 + (y^5 + 5*y^4 + 26*y^3 + 67*y^2 + 98*y + 57)*x^5 + (y^6 + 6*y^5 + 40*y^4 + 144*y^3 + 336*y^2 + 430*y + 231)*x^6 + (y^7 + 7*y^6 + 57*y^5 + 265*y^4 + 861*y^3 + 1767*y^2 + 2062*y + 1023)*x^7 + (y^8 + 8*y^7 + 77*y^6 + 440*y^5 + 1845*y^4 + 5300*y^3 + 9873*y^2 + 10610*y + 4926)*x^8 + (y^9 + 9*y^8 + 100*y^7 + 679*y^6 + 3501*y^5 + 13041*y^4 + 33974*y^3 + 58221*y^2 + 58240*y + 25483)*x^9 + (y^10 + 10*y^9 + 126*y^8 + 992*y^7 + 6083*y^6 + 27978*y^5 + 94580*y^4 + 226716*y^3 + 360930*y^2 + 338984*y + 140601)*x^10 + ...
where, by definition,
A(x,y) = Sum_{n>=0} x^n * ((1+x)^n + y)^n.
This triangle of coefficients T(n,k) of x^n*y^k in A(x,y) begins
1;
1, 1;
2, 2, 1;
5, 7, 3, 1;
16, 24, 15, 4, 1;
57, 98, 67, 26, 5, 1;
231, 430, 336, 144, 40, 6, 1;
1023, 2062, 1767, 861, 265, 57, 7, 1;
4926, 10610, 9873, 5300, 1845, 440, 77, 8, 1;
25483, 58240, 58221, 33974, 13041, 3501, 679, 100, 9, 1;
140601, 338984, 360930, 226716, 94580, 27978, 6083, 992, 126, 10, 1;
822422, 2081189, 2345469, 1572134, 706225, 226843, 54271, 9886, 1389, 155, 11, 1;
5074015, 13423258, 15926115, 11318196, 5428820, 1876728, 486941, 97448, 15246, 1880, 187, 12, 1; ...
the leftmost column in which yields A121689:
[1, 1, 2, 5, 16, 57, 231, 1023, 4926, 25483, 140601, ..., A121689, ...]
and has g.f.: Sum_{n>=0} x^n * (1+x)^(n^2).
Column 1 equals
[1, 2, 7, 24, 98, 430, 2062, 10610, 58240, 338984, ..., A325581(n), ...]
and has g.f.: Sum_{n>=0} (n+1) * x^n * (1+x)^(n*(n+1)).
Column 2 equals
[1, 3, 15, 67, 336, 1767, 9873, 58221, 360930, ..., A325586(n), ...]
and has g.f.: Sum_{n>=0} (n+1)*(n+2)/2 * x^n * (1+x)^(n*(n+2)).
The row sums of this triangle begin
[1, 2, 5, 16, 60, 254, 1188, 6043, 33080, 193249, ..., A301306(n), ...]
and has g.f.: Sum_{n>=0} (1 + (1+x)^n)^n * x^n.

Paul D. Hanna, Table of n, a(n) for n = 0..5150 terms of this triangle as read by rows 0..100

Cf. A121689 (column 0), A301306 (row sums), A325581 (column 1), A325586 (column 2), A325587 (column 3).

{T(n,k) = my(Axy = sum(m=0,n, x^m * ((1+x +x*O(x^n))^m + y)^m ) );
polcoeff( polcoeff( Axy,n,x),k,y)}
for(n=0,12,for(k=0,n, print1(T(n,k),", "));print(""))

G.f.: A(x,y) = Sum_{n>=0} x^n * Sum_{k=0..n} T(n,k)*x^n*y^k equals the following.
(1) A(x,y) = Sum_{n>=0} x^n * ((1+x)^n + y)^n.
(2) A(x,y) = Sum_{n>=0} x^n * (1+x)^(n^2) / (1 - x*y*(1+x)^n)^(n+1).
(3) A(x,y) = Sum_{k>=0} y^k * Sum_{n>=0} binomial(n+k,n) * (x*(1+x)^n)^(n+k).
G.f. of column k: Sum_{n>=0} binomial(n+k,n) * x^n * (1+x)^(n*(n+k)).

A325289 G.f. A(x) satisfies: Sum_{n>=0} x^nA(x)^(n(n+1)/2) = Sum_{n>=0} x^n*(1+x)^(n^2).

Original entry on oeis.org

1, 1, 1, 3, 9, 41, 200, 1096, 6440, 40095, 262298, 1790395, 12699751, 93311273, 708519038, 5549751855, 44780255681, 371785828813, 3173019719939, 27813799706468, 250222091088035, 2308676057468240, 21831456961064288, 211449264040904335, 2096345122112307560, 21261235260097878478, 220457711039776064974, 2335722548273384751833

Offset: 0

Paul D. Hanna, Apr 25 2019

nonn

			G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 9*x^4 + 41*x^5 + 200*x^6 + 1096*x^7 + 6440*x^8 + 40095*x^9 + 262298*x^10 + 1790395*x^11 + 12699751*x^12 + ...
such that the following series are equal:
B(x) = 1 + x*A(x) + x^2*A(x)^3 + x^3*A(x)^6 + x^4*A(x)^10 + x^5*A(x)^15 + x^6*A(x)^21 + x^7*A(x)^28 + x^8*A(x)^36 + ...
B(x) = 1 + x*(1+x) + x^2*(1+x)^4 + x^3*(1+x)^9 + x^4*(1+x)^16 + x^5*(1+x)^25 + x^6*(1+x)^36 + x^7*(1+x)^49 + x^8*(1+x)^64 + ...
where
B(x) = 1 + x + 2*x^2 + 5*x^3 + 16*x^4 + 57*x^5 + 231*x^6 + 1023*x^7 + 4926*x^8 + 25483*x^9 + 140601*x^10 + ... + A121689(n)*x^n + ...

Paul D. Hanna, Table of n, a(n) for n = 0..250

Cf. A121689, A325294, A326424.

{a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);
A[#A] = -polcoeff( sum(m=0,#A, x^m*( Ser(A)^(m*(m+1)/2) - (1+x +x*O(x^#A))^(m^2)) ),#A) );A[n+1]}
for(n=0,30,print1(a(n),", "))

A217285 Irregular triangle read by rows: T(n,k) is the number of labeled relations on n nodes with exactly k edges; n>=0, 0<=k<=n^2.

Original entry on oeis.org

1, 1, 1, 1, 4, 6, 4, 1, 1, 9, 36, 84, 126, 126, 84, 36, 9, 1, 1, 16, 120, 560, 1820, 4368, 8008, 11440, 12870, 11440, 8008, 4368, 1820, 560, 120, 16, 1, 1, 25, 300, 2300, 12650, 53130, 177100, 480700, 1081575, 2042975, 3268760, 4457400, 5200300, 5200300, 4457400, 3268760, 2042975, 1081575, 480700, 177100, 53130, 12650, 2300, 300, 25, 1

Offset: 0

Geoffrey Critzer, Sep 30 2012

A labeled relation on 6 nodes will be connected with probability > 99%. It will have at least 10 and no more than 26 edges with probability > 99%.
A random labeled relation can be generated in Mathematica:
GraphPlot[g=Table[RandomInteger[],{6},{6}], DirectedEdges->True, VertexLabeling->True, SelfLoopStyle->True, MultiedgeStyle->True]
Sum {k=0...n^2} T(n,k)*k = A185968. - Geoffrey Critzer, Oct 07 2012

			G.f.: A(x,y) = 1 + x*(1+y) + x^2*(1+y)^4 + x^3*(1+y)^9 + x^4*(1+y)^16 +...
Triangle T(n,k) begins:
1;
1,  1;
1,  4,  6,    4,    1;
1,  9,  36,  84,  126,  126,   84,    36,     9,     1;
1, 16, 120, 560, 1820, 4368, 8008, 11440, 12870, 11440, ...

Paul D. Hanna, Rows 0..20, as a flattened table of n, a(n) for n = 0..2890.

Column k=1 gives: A000290.
Row lengths are: A002522.
Antidiagonal sums: A121689.

Table[Table[Binomial[n^2,k], {k,0,n^2}], {n,0,6}] //Grid

{T(n,k)=polcoeff((1+x+x*O(x^k))^(n^2),k)}
for(n=0,6,for(k=0,n^2,print1(T(n,k),", "));print("")) \\ Paul D. Hanna, Aug 22 2013

{T(n,k)=polcoeff(polcoeff(sum(m=0, n, x^m*(1+y)^m*prod(k=1, m, (1-x*(1+y)^(4*k-3))/(1-x*(1+y)^(4*k-1) +x*O(x^n)))), n,x),k,y)}
{for(n=0,6,for(k=0,n^2,print1(T(n,k),", "));print(""))} \\ Paul D. Hanna, Aug 22 2013

T(n,k) = binomial(n^2,k).
E.g.f.: Sum{n>=0}(1+y)^(n^2)*x^n/n!. - Geoffrey Critzer, Oct 07 2012
G.f.: A(x,y) = Sum_{n>=0} x^n*(1+y)^n*Product_{k=1..n} (1-x*(1+y)^(4*k-3))/(1-x*(1+y)^(4*k-1)) due to a q-series identity. - Paul D. Hanna, Aug 22 2013
G.f.: A(x,y) = 1/(1- q*x/(1- (q^3-q)*x/(1- q^5*x/(1- (q^7-q^3)*x/(1- q^9*x/(1- (q^11-q^5)*x/(1- q^13*x/(1- (q^15-q^7)*x/(1- ...))))))))), a continued fraction where q = (1+y), due to an identity of a partial elliptic theta function. - Paul D. Hanna, Aug 22 2013

A325581 G.f.: Sum_{n>=0} (n+1) * x^n * (1+x)^(n*(n+1)).

Original entry on oeis.org

1, 2, 7, 24, 98, 430, 2062, 10610, 58240, 338984, 2081189, 13423258, 90626012, 638509008, 4682120763, 35650040782, 281266115870, 2295142774336, 19338107378888, 167987656339604, 1502475101768767, 13818574571596432, 130542011977462175, 1265358001625542030, 12572822521590475349, 127943980062492526520, 1332336499429857507073, 14186629118985647254622, 154348478009342665050329, 1714707987491310848285920

Offset: 0

Paul D. Hanna, May 11 2019

nonn

Equals column 1 of triangle A325580.

			G.f.: A(x) = 1 + 2*x + 7*x^2 + 24*x^3 + 98*x^4 + 430*x^5 + 2062*x^6 + 10610*x^7 + 58240*x^8 + 338984*x^9 + 2081189*x^10 + 13423258*x^11 + 90626012*x^12 + ...
such that
A(x) = 1 + 2*x*(1+x)^2 + 3*x^2*(1+x)^6 + 4*x^3*(1+x)^12 + 5*x^4*(1+x)^20 + 6*x^5*(1+x)^30 + 7*x^6*(1+x)^42 + 8*x^7*(1+x)^(56) + 9*x^8*(1+x)^72 + ...

Cf. A325580, A121689, A325586, A325587.

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

A325586 G.f.: Sum_{n>=0} (n+1)(n+2)/2 x^n * (1+x)^(n*(n+2)).

Original entry on oeis.org

1, 3, 15, 67, 336, 1767, 9873, 58221, 360930, 2345469, 15926115, 112702725, 829218143, 6329731749, 50032666719, 408810685879, 3447546750090, 29963861568735, 268051909321565, 2465213070499965, 23282355990573738, 225577403162464915, 2240023319131286013, 22778185448591006709, 236997065442660095669, 2521130509681288754841, 27401150807636634911205, 304071227823781106763523, 3443058535424619400592874

Offset: 0

Paul D. Hanna, May 11 2019

nonn

Equals column 2 of triangle A325580.

			G.f.: A(x) = 1 + 3*x + 15*x^2 + 67*x^3 + 336*x^4 + 1767*x^5 + 9873*x^6 + 58221*x^7 + 360930*x^8 + 2345469*x^9 + 15926115*x^10 + 112702725*x^11 + ...
such that
A(x) = 1 + 3*x*(1+x)^3 + 6*x^2*(1+x)^8 + 10*x^3*(1+x)^15 + 15*x^4*(1+x)^24 + 21*x^5*(1+x)^35 + 28*x^6*(1+x)^48 + 36*x^7*(1+x)^63 + 45*x^8*(1+x)^80 + ...

Cf. A325580, A121689, A325581, A325587.

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

A320950 G.f.: [ Sum_{n>=0} x^n * (1+x)^(n^2) ] * [ Sum_{n>=0} x^n / (1+x)^(n^2) ].

Original entry on oeis.org

1, 2, 3, 5, 20, 81, 272, 1144, 6147, 30859, 158137, 955988, 5995439, 37307475, 252176301, 1813873656, 13149151909, 99412177075, 793516947530, 6470733413532, 54217400538306, 473499984230701, 4245890615280401, 38948094201082823, 368815668052736968, 3585473523132486254, 35608100771085923165, 362850695679003347638, 3788143752503214124895

Offset: 0

Paul D. Hanna, Oct 26 2018

nonn

			G.f.: A(x) = 1 + 2*x + 3*x^2 + 5*x^3 + 20*x^4 + 81*x^5 + 272*x^6 + 1144*x^7 + 6147*x^8 + 30859*x^9 + 158137*x^10 + 955988*x^11 + 5995439*x^12 + ...
such that A(x) = P(x) * Q(x) where
P(x) = 1 + x*(1+x) + x^2*(1+x)^4 + x^3*(1+x)^9 + x^4*(1+x)^16 + x^5*(1+x)^25 + x^6*(1+x)^36 + x^7*(1+x)^49 + ... + x^n * (1+x)^(n^2) + ...
Q(x) = 1 + x/(1+x) + x^2/(1+x)^4 + x^3/(1+x)^9 + x^4/(1+x)^16 + x^5/(1+x)^25 + x^6/(1+x)^36 + x^7/(1+x)^49 + ... + x^n / (1+x)^(n^2) + ...
Explicitly,
P(x) = 1 + x + 2*x^2 + 5*x^3 + 16*x^4 + 57*x^5 + 231*x^6 + 1023*x^7 + 4926*x^8 + 25483*x^9 + 140601*x^10 + 822422*x^11 + ... + A121689(n)*x^n + ...
Q(x) = 1 + x - 2*x^3 + x^4 + 11*x^5 - 19*x^6 - 86*x^7 + 365*x^8 + 581*x^9 - 7336*x^10 + 6061*x^11 + 142946*x^12 - 556061*x^13 + ...

Cf. A121689, A320830.

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

A337850 G.f. A(x) satisfies: Sum_{n>=0} x^(n^2) * A(x)^n = Sum_{n>=0} x^n * (1+x)^(n^2).

Original entry on oeis.org

1, 2, 5, 15, 53, 217, 973, 4735, 24686, 137026, 805273, 4986541, 32409056, 220327688, 1562196894, 11522725486, 88215618320, 699573288808, 5736354452771, 48556636776122, 423674461139747, 3805446588538974, 35142868684678717, 333303148345306269, 3243121812554272131

Offset: 0

Paul D. Hanna, Sep 26 2020

nonn

			G.f.: A(x) = 1 + 2*x + 5*x^2 + 15*x^3 + 53*x^4 + 217*x^5 + 973*x^6 + 4735*x^7 + 24686*x^8 + 137026*x^9 + 805273*x^10 + ...
such that the following series are equal
B(x) = 1 + x*A(x) + x^4*A(x)^2 + x^9*A(x)^3 + x^16*A(x)^4 + x^25*A(x)^5 + x^36*A(x)^6 + x^49*A(x)^7 + ... + x^(n^2)*A(x)^n + ...
B(x) = 1 + x*(1+x) + x^2*(1+x)^4 + x^3*(1+x)^9 + x^4*(1+x)^16 + x^5*(1+x)^25 + x^6*(1+x)^36 + x^7*(1+x)^49 + ... + x^n*(1+x)^(n^2) + ...
where
B(x) = 1 + x + 2*x^2 + 5*x^3 + 16*x^4 + 57*x^5 + 231*x^6 + 1023*x^7 + 4926*x^8 + 25483*x^9 + 140601*x^10 + ... + A121689(n)*x^n + ...

Paul D. Hanna, Table of n, a(n) for n = 0..200

Cf. A325298, A121689.

{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = -polcoeff( sum(m=0, #A, x^(m^2)*Ser(A)^m - x^m*(1+x +x*O(x^#A) )^(m^2) ), #A) ); A[n+1]}
for(n=0, 35, print1(a(n), ", "))

A183608 G.f.: A(x) = Sum_{n>=0} x^n * C(x)^(n^2), where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).

Original entry on oeis.org

1, 1, 2, 7, 29, 133, 658, 3471, 19400, 114417, 709815, 4619048, 31446579, 223419752, 1652599036, 12698380493, 101151995810, 833740791381, 7098646227614, 62335051895044, 563749889969108, 5244173616702347, 50117689766439784

Offset: 0

Paul D. Hanna, Jan 15 2011

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 29*x^4 + 133*x^5 + 658*x^6 +...

Cf. A121689, A000108.

Flatten[{1,Table[Sum[Binomial[(n-k)^2+2*k, k] * (n-k)^2/((n-k)^2 + 2*k),{k,0,n-1}],{n,1,20}]}] (* Vaclav Kotesovec, Mar 06 2014 *)

{a(n)=if(n<0,0,0^n+sum(k=0, n-1, binomial((n-k)^2+2*k, k)*(n-k)^2/((n-k)^2+2*k)))}

a(n) = Sum_{k=0..n-1} binomial((n-k)^2+2k, k) * (n-k)^2/((n-k)^2 + 2k) for n>0 with a(0)=1.
G.f.: A(x) = Sum_{n>=0} x^n*C(x)^n*Product_{k=1..n} (1-x*C(x)^(4*k-3))/(1-x*C(x)^(4*k-1)) where C(x) = 1 + x*C(x)^2.
Let q = C(x) = 1 + x*C(x)^2, then g.f. A(x) equals the continued fraction:
A(x) = 1/(1- q*x/(1- q*(q^2-1)*x/(1- q^5*x/(1- q^3*(q^4-1)*x/(1- q^9*x/(1- q^5*(q^6-1)*x/(1- q^13*x/(1- q^7*(q^8-1)*x/(1- ...)))))))))
due to an identity of a partial elliptic theta function.
G.f.: A(x) = 1 + x*C(x)* G( x*C(x)^2 ), where G(x) = Sum_{k>=0} x^k*(1+x)^(k^2) is the g.f. of A121689.

cp's OEIS Frontend

A121690 G.f.: A(x) = Sum_{k>=0} x^k * (1+x)^(k*(k+1)/2).

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A325580 G.f.: A(x,y) = Sum_{n>=0} x^n * ((1+x)^n + y)^n, where A(0) = 0, as a triangle of coefficients T(n,k) of x^n*y^k in A(x,y) = Sum_{n>=0} x^n * Sum_{k=0..n} T(n,k)*x^n*y^k, read by rows.

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PARI

Formula

A325289 G.f. A(x) satisfies: Sum_{n>=0} x^n*A(x)^(n*(n+1)/2) = Sum_{n>=0} x^n*(1+x)^(n^2).

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PARI

A217285 Irregular triangle read by rows: T(n,k) is the number of labeled relations on n nodes with exactly k edges; n>=0, 0<=k<=n^2.

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Mathematica

PARI

PARI

Formula

A325581 G.f.: Sum_{n>=0} (n+1) * x^n * (1+x)^(n*(n+1)).

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PARI

A325586 G.f.: Sum_{n>=0} (n+1)*(n+2)/2 * x^n * (1+x)^(n*(n+2)).

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PARI

A320950 G.f.: [ Sum_{n>=0} x^n * (1+x)^(n^2) ] * [ Sum_{n>=0} x^n / (1+x)^(n^2) ].

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PARI

A337850 G.f. A(x) satisfies: Sum_{n>=0} x^(n^2) * A(x)^n = Sum_{n>=0} x^n * (1+x)^(n^2).

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PARI

A325580 G.f.: A(x,y) = Sum_{n>=0} x^n * ((1+x)^n + y)^n, where A(0) = 0, as a triangle of coefficients T(n,k) of x^ny^k in A(x,y) = Sum_{n>=0} x^n Sum_{k=0..n} T(n,k)x^ny^k, read by rows.

A325289 G.f. A(x) satisfies: Sum_{n>=0} x^nA(x)^(n(n+1)/2) = Sum_{n>=0} x^n*(1+x)^(n^2).

A325586 G.f.: Sum_{n>=0} (n+1)(n+2)/2 x^n * (1+x)^(n*(n+2)).