A122400 -id:A122400 - cp's OEIS frontend

A220353 G.f.: Sum_{n>=0} (1 - (1-x)^n)^n.

1, 1, 4, 23, 176, 1697, 19805, 271669, 4285195, 76430799, 1521161530, 33422603485, 803584699252, 20986514811397, 591616582807036, 17905570068475471, 579092313210791549, 19931241131544637637, 727395001560116046739, 28057672464546863483509, 1140566596105346550309751, 48735378037084078566334897, 2183719157723179429519093520, 102386962560815561519635957007

Offset: 0

Paul D. Hanna, Dec 11 2012

nonn

Limit n->infinity A220353(n)/A187826(n) = 1. - Vaclav Kotesovec, Nov 08 2014

			G.f.: A(x) = 1 + x + 4*x^2 + 23*x^3 + 176*x^4 + 1697*x^5 + 19805*x^6 +...
where the g.f. satisfies the identities:
(1) A(x) = 1 + x + (2*x - x^2)^2 + (3*x - 3*x^2 + x^3)^3 + (4*x - 6*x^2 + 4*x^3 - x^4)^4 + (5*x - 10*x^2 + 10*x^3 - 5*x^4 + x^5)^5 +...
(2) A(x) = (1-x) + (1-x)^2*(2*x - x^2) + (1-x)^3*(3*x - 3*x^2 + x^3)^2 + (1-x)^4*(4*x - 6*x^2 + 4*x^3 - x^4)^3 + (1-x)^5*(5*x - 10*x^2 + 10*x^3 - 5*x^4 + x^5)^4 +...

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

Cf. A220352, A122400, A187827, A187826.

terms = 24;
gf = 1 + Sum[(1 - (1 - x)^n)^n, {n, 1, terms}] + O[x]^terms;
CoefficientList[gf, x] (* Jean-François Alcover, Jul 01 2018 *)

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

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

G.f.: Sum_{n>=1} (1-x)^n * (1 - (1-x)^n)^(n-1).
a(n) = c * n! / (sqrt(n) * (log(2))^(2*n)), where c = 0.93418651575946259471737... . - Vaclav Kotesovec, May 06 2014
In closed form, c = 2^(log(2)/2-1) / (log(2) * sqrt(Pi*(1-log(2)))). - Vaclav Kotesovec, May 03 2015

a(22)-a(23) corrected by Andrew Howroyd, Feb 22 2018

A301584 G.f.: Sum_{n>=0} ((1+x)^(2*n) - 1)^n.

Original entry on oeis.org

1, 2, 17, 264, 5784, 163610, 5667551, 232280480, 10991951114, 589780778314, 35379149504709, 2346218124687516, 170439977706143335, 13459938431949414118, 1148107512505151099653, 105194122765096703619248, 10303686044959088279454117, 1074408525677705370497704526, 118828297870115694372235974855, 13893778686151373846512389392672, 1712370237144948501135060958863978

Offset: 0

Paul D. Hanna, Mar 24 2018

nonn

			G.f.: A(x) = 1 + 2*x + 17*x^2 + 264*x^3 + 5784*x^4 + 163610*x^5 + 5667551*x^6 + 232280480*x^7 + 10991951114*x^8 + 589780778314*x^9 + ...
such that
A(x) = 1 + ((1+x)^2-1) + ((1+x)^4-1)^2 + ((1+x)^6-1)^3 + ((1+x)^8-1)^4 + ((1+x)^10-1)^5 + ((1+x)^12-1)^6 + ((1+x)^14-1)^7 + ...
Also,
A(x) = 1/2 + (1+x)^2/(1 + (1+x)^2)^2 + (1+x)^8/(1 + (1+x)^4)^3 + (1+x)^18/(1 + (1+x)^6)^4 + (1+x)^32/(1 + (1+x)^8)^5 + (1+x)^50/(1 + (1+x)^10)^6 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..340

Cf. A122400, A301585, A301586.

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

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

a(n) ~ c * d^n * n! / sqrt(n), where d = 2*A317855 = 6.3221773077308576276603444051762649834527655483771126832545564150753941184386... and c = 0.302715376391132275494451399946850989516917... - Vaclav Kotesovec, Aug 09 2018

A302615 G.f.: Sum_{n>=0} (3 + (1+x)^n)^n / (4 + (1+x)^n)^(n+1).

Original entry on oeis.org

1, 1, 10, 130, 2390, 56714, 1644138, 56327820, 2226708772, 99761490536, 4995375316146, 276464859358474, 16757956600528786, 1104116777798713154, 78565751676021256606, 6004629888868350015506, 490572645247461234631946, 42665124626946741636482996, 3935474733572880332326074450, 383756013888633346483785849474

Offset: 0

Paul D. Hanna, Apr 10 2018

nonn

The following identity holds for |y| <= 1 and fixed real k > 0:

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

			G.f.: A(x) = 1 + x + 10*x^2 + 130*x^3 + 2390*x^4 + 56714*x^5 + 1644138*x^6 + 56327820*x^7 + 2226708772*x^8 + 99761490536*x^9 + ...
such that
A(x) = 1/5  +  (3 + (1+x))/(4 + (1+x))^2  +  (3 + (1+x)^2)^2/(4 + (1+x)^2)^3  +  (3 + (1+x)^3)^3/(4 + (1+x)^3)^4  +  (3 + (1+x)^4)^4/(4 + (1+x)^4)^5  +  (3 + (1+x)^5)^5/(4 + (1+x)^5)^6  +  (3 + (1+x)^6)^6/(4 + (1+x)^6)^7  + ...
Also,
A(x) = 1  +  ((1+x) - 1)/(4 - 3*(1+x))^2  +  ((1+x)^2 - 1)^2/(4 - 3*(1+x)^2)^3  +  ((1+x)^3 - 1)^3/(4 - 3*(1+x)^3)^4  +  ((1+x)^4 - 1)^4/(4 - 3*(1+x)^4)^5  +  ((1+x)^5 - 1)^5/(4 - 3*(1+x)^5)^6  +  ((1+x)^6 - 1)^6/(4 - 3*(1+x)^6)^7 + ...

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

Cf. A122400, A302598, A302614.

{a(n) = my(A=1, o=x*O(x^n)); A = sum(m=0, n, ((1+x +o)^m - 1)^m / (4 - 3*(1+x +o)^m)^(m+1)); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

G.f.: Sum_{n>=0} ((1+x)^n - 1)^n / (4 - 3*(1+x)^n)^(n+1).

a(n) ~ c * d^n * n! / sqrt(n), where d = 5.2709551504518355656831902094014170087... and c = 0.26621450180820822374221893929... - Vaclav Kotesovec, Aug 09 2018

A122419 Number of labeled digraphs with n arcs and with no vertex of indegree 0.

Original entry on oeis.org

1, 0, 1, 8, 93, 1354, 23900, 496244, 11855700, 320428318, 9667220397, 322072882348, 11744421711587, 465270864839688, 19899234175413257, 913836170567749048, 44849438199960187278, 2342666125012348876152

Offset: 0

Vladeta Jovovic, Sep 03 2006

Cf. A122420, A122400.

A122418 := proc(n) option remember ; add( combinat[stirling2](n,k)*(k-1)^n*k!,k=0..n) ; end: A122419 := proc(n) option remember ; add( combinat[stirling1](n,k)*A122418(k),k=0..n)/n! ; end: for n from 0 to 30 do printf("%d, ",A122419(n)) ; od ; # R. J. Mathar, May 18 2007

nmax=20; CoefficientList[Series[Sum[((1+x)^(n-1)-1)^n, {n, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, May 06 2014 *)

a(n) = (1/n!)*Sum_{k=0..n} Stirling1(n,k)*A122418(k).
G.f.: Sum_{n>=0} ((1+x)^(n-1) - 1)^n.
a(n) ~ c * d^n * n! / sqrt(n), where d = A317855 = (1+exp(1/r))*r^2 = 3.161088653865428813830172202588132491726382774188556341627278..., r = 0.8737024332396683304965683047207192982139922672025395099... is the root of the equation exp(1/r)/r + (1+exp(1/r))*LambertW(-exp(-1/r)/r) = 0, and c = 0.08904589343883135100956914504938... . - Vaclav Kotesovec, May 07 2014

More terms from R. J. Mathar, May 18 2007

A301585 G.f.: Sum_{n>=0} ((1+x)^(3*n) - 1)^n.

Original entry on oeis.org

1, 3, 39, 910, 29949, 1271751, 66116065, 4066082856, 288701376912, 23240635243591, 2091554595246705, 208085119389952134, 22676957610808295192, 2686515300821612112411, 343760257348413122290260, 47248346582443326267328400, 6942339982115290619799947901, 1085919469129099832397573088863, 180160797497273341662653292624309, 31598815412054398239059538582525618

Offset: 0

Paul D. Hanna, Mar 24 2018

nonn

			G.f.: A(x) = 1 + 3*x + 39*x^2 + 910*x^3 + 29949*x^4 + 1271751*x^5 + 66116065*x^6 + 4066082856*x^7 + 288701376912*x^8 + ...
such that
A(x) = 1 + ((1+x)^3-1) + ((1+x)^6-1)^2 + ((1+x)^9-1)^3 + ((1+x)^12-1)^4 + ((1+x)^15-1)^5 + ((1+x)^18-1)^6 + ((1+x)^21-1)^7 + ...
Also,
A(x) = 1/2 + (1+x)^3/(1 + (1+x)^3)^2 + (1+x)^12/(1 + (1+x)^6)^3 + (1+x)^27/(1 + (1+x)^9)^4 + (1+x)^48/(1 + (1+x)^12)^5 + (1+x)^75/(1 + (1+x)^15)^6 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..325

Cf. A122400, A301584, A301586.

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

G.f.: Sum_{n>=0} (1+x)^(3*n^2) /(1 + (1+x)^(3*n))^(n+1).

a(n) ~ c * d^n * n! / sqrt(n), where d = 3*A317855 = 9.4832659615962864414905166077643974751791483225656690248818346226130911776579... and c = 0.3108017465925995208675813879173750641359609... - Vaclav Kotesovec, Aug 09 2018

A301586 G.f.: Sum_{n>=0} ((1+x)^(4*n) - 1)^n.

Original entry on oeis.org

1, 4, 70, 2180, 95729, 5422192, 375951144, 30833206304, 2919367902648, 313380517364324, 37606931999739230, 4988933437333555060, 724960700435104219679, 114519163835687116024256, 19538926882901715534673728, 3580844611314789257667535968, 701546780854024941112271649610, 146318317830136401429653726419700, 32367591848747955557013839920695374, 7569528177000020896435962191564396740

Offset: 0

Paul D. Hanna, Mar 24 2018

nonn

			G.f.: A(x) = 1 + 4*x + 70*x^2 + 2180*x^3 + 95729*x^4 + 5422192*x^5 + 375951144*x^6 + 30833206304*x^7 + ...
such that
A(x) = 1 + ((1+x)^4-1) + ((1+x)^8-1)^2 + ((1+x)^12-1)^3 + ((1+x)^16-1)^4 + ((1+x)^20-1)^5 + ((1+x)^24-1)^6 + ((1+x)^28-1)^7 + ...
Also,
A(x) = 1/2 + (1+x)^4/(1 + (1+x)^4)^2 + (1+x)^16/(1 + (1+x)^8)^3 + (1+x)^36/(1 + (1+x)^12)^4 + (1+x)^64/(1 + (1+x)^16)^5 + (1+x)^100/(1 + (1+x)^20)^6 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..314

Cf. A122400, A301584, A301585.

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

G.f.: Sum_{n>=0} (1+x)^(4*n^2) /(1 + (1+x)^(4*n))^(n+1).

a(n) ~ c * d^n * n! / sqrt(n), where d = 4*A317855 = 12.64435461546171525532068881035252996690553109675422536650911283015078823687... and c = 0.31492557816516652573983016205911709623053... - Vaclav Kotesovec, Aug 09 2018

A302614 G.f.: Sum_{n>=0} (2 + (1+x)^n)^n / (3 + (1+x)^n)^(n+1).

Original entry on oeis.org

1, 1, 8, 91, 1474, 30765, 785053, 23682833, 824522797, 32537599175, 1435199414014, 69973425937141, 3736662443907962, 216901789032691605, 13598124265965160130, 915670842666691879191, 65913110467411283181409, 5050836914009172555862713, 410501468976427335127369669, 35269929119728622895198302033, 3194195105084750546987502710855

Offset: 0

Paul D. Hanna, Apr 10 2018

nonn

The following identity holds for |y| <= 1 and fixed real k > 0:

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

			G.f.: A(x) = 1 + x + 8*x^2 + 91*x^3 + 1474*x^4 + 30765*x^5 + 785053*x^6 + 23682833*x^7 + 824522797*x^8 + 32537599175*x^9 + ...
such that
A(x) = 1/4  +  (2 + (1+x))/(3 + (1+x))^2  +  (2 + (1+x)^2)^2/(3 + (1+x)^2)^3  +  (2 + (1+x)^3)^3/(3 + (1+x)^3)^4  +  (2 + (1+x)^4)^4/(3 + (1+x)^4)^5  +  (2 + (1+x)^5)^5/(3 + (1+x)^5)^6  +  (2 + (1+x)^6)^6/(3 + (1+x)^6)^7  + ...
Also,
A(x) = 1  +  ((1+x) - 1)/(3 - 2*(1+x))^2  +  ((1+x)^2 - 1)^2/(3 - 2*(1+x)^2)^3  +  ((1+x)^3 - 1)^3/(3 - 2*(1+x)^3)^4  +  ((1+x)^4 - 1)^4/(3 - 2*(1+x)^4)^5  +  ((1+x)^5 - 1)^5/(3 - 2*(1+x)^5)^6  +  ((1+x)^6 - 1)^6/(3 - 2*(1+x)^6)^7 + ...
RELATED INFINITE SERIES.
(1) At x = -1/3: the following sums are equal
S1 = Sum_{n>=0} 3^n * (2*3^n + 2^n)^n / (3^(n+1) + 2^n)^(n+1),
S1 = Sum_{n>=0} (-3)^n * (3^n - 2^n)^n / (3^(n+1) - 2^(n+1))^(n+1).
Explicitly,
S1 = 1/4 + 3*8/11^2 + 9*22^2/31^3 + 27*62^3/89^4 + 81*178^4/259^5 + 243*518^5/761^6 + 729*1522^6/2251^7 + 2187*4502^7/6689^8 + 6561*13378^8/19939^9 + 19683*39878^9/59561^10 + ...
S1 = 1 - 3*1/5^2 + 9*5^2/19^3 - 27*19^3/65^4 + 81*65^4/211^5 - 243*211^5/665^6 + 729*665^6/2059^7 - 2187*2059^7/6305^8 + 6561*6305^8/19171^9 - 19683*19171^9/58025^10 + ...
where S1 = 0.90501051059439877583104471171480036033530856741889530664913...
(2) At x = -1/2: the following sums are equal
S2 = Sum_{n>=0} 2^n * (2^(n+1) + 1)^n / (3*2^n + 1)^(n+1),
S2 = Sum_{n>=0} (-2)^n * (2^n - 1)^n / (3*2^n - 2)^(n+1).
Explicitly,
S2 = 1/4 + 2*5/7^2 + 4*9^2/13^3 + 8*17^3/25^4 + 16*33^4/49^5 + 32*65^5/97^6 + 64*129^6/193^7 + 128*257^7/385^8 + 256*513^8/769^9 + 512*1025^9/1537^10 + ...
S2 = 1 - 2*1/4^2 + 4*3^2/10^3 - 8*7^3/22^4 + 16*15^4/46^5 - 32*31^5/94^6 + 64*63^6/190^7 - 128*127^7/382^8 + 256*255^8/766^9 - 512*511^9/1534^10 + ...
where S2 = 0.90222608896798122564942421232120719521782835530371831680447...

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

Cf. A122400, A302598, A302615.

{a(n) = my(A=1, o=x*O(x^n)); A = sum(m=0, n, ((1+x +o)^m - 1)^m / (3 - 2*(1+x +o)^m)^(m+1)); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

G.f.: Sum_{n>=0} ((1+x)^n - 1)^n / (3 - 2*(1+x)^n)^(n+1).

a(n) ~ c * d^n * n! / sqrt(n), where d = 4.64471605501103711823541367464... and c = 0.270134222044915506270113032... - Vaclav Kotesovec, Aug 10 2018

A227619 G.f.: A(x) = 1+x + Sum_{n>=2} (A(x)^n - 1)^n.

Original entry on oeis.org

1, 1, 4, 63, 1278, 29764, 758065, 20611793, 590579518, 17707907024, 553879330720, 18066513887790, 615744470668778, 22014659625607877, 830262409494773896, 33243718957578687811, 1422095813097928147636, 65311403344808947050730, 3227884786251446164710376

Offset: 0

Paul D. Hanna, Aug 21 2013

nonn

			G.f.: A(x) = 1 + x + 4*x^2 + 63*x^3 + 1278*x^4 + 29764*x^5 +...
where
A(x) = 1+x + (A(x)^2 - 1)^2 + (A(x)^3 - 1)^3 + (A(x)^4 - 1)^4 + (A(x)^5 - 1)^5 +...

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

Cf. A122400.

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

a(n) ~ c * d^n * n! / sqrt(n), where d = A317855 = (1+exp(1/r))*r^2 = 3.161088653865428813830172202588132491726382774188556341627278..., r = 0.8737024332396683304965683047207192982139922672025395099... is the root of the equation exp(1/r)/r + (1+exp(1/r))*LambertW(-exp(-1/r)/r) = 0, and c = 0.9913753087... . - Vaclav Kotesovec, May 07 2014

A232192 G.f. satisfies: A(x) = 1 + x*Sum_{n>=0} (A(x)^n - 1)^n.

Original entry on oeis.org

1, 1, 1, 5, 44, 519, 7590, 132347, 2689046, 62644234, 1651650774, 48731341965, 1592908456996, 57173688136781, 2235773294509565, 94608603077007214, 4306708055122614542, 209823573154587335730, 10892496561736261641371, 600171728539156939466278

Offset: 0

Paul D. Hanna, Nov 20 2013

nonn

			G.f.: A(x) = 1 + x + x^2 + 5*x^3 + 44*x^4 + 519*x^5 + 7590*x^6 + 132347*x^7 + 2689046*x^8 + 62644234*x^9 + 1651650774*x^10 +...
where
A(x) = 1 + x + x*(A(x)-1) + x*(A(x)^2-1)^2 + x*(A(x)^3-1)^3 + x*(A(x)^4-1)^4 + x*(A(x)^5-1)^5 + x*(A(x)^6-1)^6 + x*(A(x)^7-1)^7 +...
Also,
A(x) = 1 + x/2  +  x*A(x)/(1 + A(x))^2  +  x*A(x)^4/(1 + A(x)^2)^3  +  x*A(x)^9/(1 + A(x)^3)^4  +  x*A(x)^16/(1 + A(x)^4)^5  +  x*A(x)^25/(1 + A(x)^5)^6  + ...

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

Cf. A122400, A192945, A192946, A192947, A192948.

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

G.f. satisfies:
(1) A(x) = 1 + x*Sum_{n>=0} A(x)^(n^2) / (1 + A(x)^n)^(n+1). - Paul D. Hanna, Mar 31 2018
(2) A(x) = 1 + Series_Reversion(x/G(x))
(3) A(x) = 1 + x*G(A(x)-1)
where G(x) is the g.f. of A122400, the number of square (0,1)-matrices without zero rows and with exactly n entries equal to 1.
a(n) ~ c * d^n * n! / n^(3/2), where d = A317855 = (1+exp(1/r))*r^2 = 3.161088653865428813830172202588132491726382774188556341627278..., r = 0.8737024332396683304965683047207192982139922672025395099... is the root of the equation exp(1/r)/r + (1+exp(1/r))*LambertW(-exp(-1/r)/r) = 0, and c = 0.12140554666... . - Vaclav Kotesovec, May 07 2014

A317798 G.f.: Sum_{n>=0} (3*(1+x)^n - 1)^n / 3^(n+1).

Original entry on oeis.org

1, 15, 786, 69261, 8554530, 1359020643, 263929299177, 60582032629791, 16046282916588207, 4817035600778756553, 1616224504900354928832, 599373591433178971787007, 243449152911402772344286998, 107482020677618238226506065235, 51249638236281451846248205583562, 26247197050200652206165329786055981, 14369481728948627418149559363836673273

Offset: 0

Paul D. Hanna, Aug 14 2018

nonn

			G.f.: A(x) = 1 + 15*x + 786*x^2 + 69261*x^3 + 8554530*x^4 + 1359020643*x^5 + 263929299177*x^6 + 60582032629791*x^7 + 16046282916588207*x^8 + ...
such that
A(x) = 1/3  +  (3*(1+x) - 1)/3^2  +  (3*(1+x)^2 - 1)^3/3^3  +  (3*(1+x)^3 - 1)^3/3^4  +  (3*(1+x)^4 - 1)^4/3^5  +  (3*(1+x)^5 - 1)^5/3^6  + ...
Also,
A(x) = 1/4  +  3*(1+x)/(3 + (1+x))^2  +  3^2*(1+x)^4/(3 + (1+x)^2)^3  +  3^3*(1+x)^9/(3 + (1+x)^3)^4  +  3^4*(1+x)^16/(3 + (1+x)^4)^5  +  3^5*(1+x)^25/(3 + (1+x)^5)^6  +  3^6*(1+x)^36/(3 + (1+x)^6)^7  + ...

Cf. A122400, A301463, A317799, A301582.

G.f. satisfies:
(1) Sum_{n>=0} 3^n * (1+x)^(n^2) / (3 + (1+x)^n)^(n+1).
(2) Sum_{n>=0} ((1+x)^n - 1/3)^n / 3.

cp's OEIS Frontend

A220353 G.f.: Sum_{n>=0} (1 - (1-x)^n)^n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

A301584 G.f.: Sum_{n>=0} ((1+x)^(2*n) - 1)^n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A302615 G.f.: Sum_{n>=0} (3 + (1+x)^n)^n / (4 + (1+x)^n)^(n+1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A122419 Number of labeled digraphs with n arcs and with no vertex of indegree 0.

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A301585 G.f.: Sum_{n>=0} ((1+x)^(3*n) - 1)^n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A301586 G.f.: Sum_{n>=0} ((1+x)^(4*n) - 1)^n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A302614 G.f.: Sum_{n>=0} (2 + (1+x)^n)^n / (3 + (1+x)^n)^(n+1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A227619 G.f.: A(x) = 1+x + Sum_{n>=2} (A(x)^n - 1)^n.

Views

Author