A380178
Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of e.g.f. B(x)^k, where B(x) is the e.g.f. of A162659.
Original entry on oeis.org
1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 8, 22, 0, 1, 4, 15, 62, 281, 0, 1, 5, 24, 126, 792, 5396, 0, 1, 6, 35, 220, 1641, 14922, 142297, 0, 1, 7, 48, 350, 2960, 30708, 384316, 4865806, 0, 1, 8, 63, 522, 4905, 55604, 777537, 12836406, 207407489, 0, 1, 9, 80, 742, 7656, 93300, 1393720, 25450806, 535396784, 10710044776, 0
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, 6, ...
0, 3, 8, 15, 24, 35, 48, ...
0, 22, 62, 126, 220, 350, 522, ...
0, 281, 792, 1641, 2960, 4905, 7656, ...
0, 5396, 14922, 30708, 55604, 93300, 148446, ...
0, 142297, 384316, 777537, 1393720, 2330305, 3716532, ...
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a(n, k) = if(k==0, 0^n, k*sum(j=0, n, (n-j+k)^(j-1)*binomial(n, j)*a(n-j, j)));
A162695
E.g.f. satisfies A(x) = exp( x*A(x) * exp(x*A(x)) ).
Original entry on oeis.org
1, 1, 5, 43, 549, 9341, 199303, 5122503, 154174121, 5321093689, 207228932811, 8991136486619, 430126003707997, 22494400020052533, 1276807091011902479, 78178242047074260751, 5136433584083525179857, 360458257425576984629873
Offset: 0
E.g.f.: A(x) = 1 + x + 5*x^2/2! + 43*x^3/3! + 549*x^4/4! + 9341*x^5/5! +...
exp(x*A(x)) = 1 + x + 3*x^2/2! + 22*x^3/3! + 257*x^4/4! + 4136*x^5/5! +...
Log(A(x)) = x + 4*x^2/2! + 30*x^3/3! + 356*x^4/4! + 5780*x^5/5! +...;
compare log(A(x)) to the e.g.f. of A055779 given by:
x + 2*x^2/2! + 10*x^3/3! + 89*x^4/4! + 1156*x^5/5! +...
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Flatten[{1,Table[Sum[Binomial[n,k] * (n+1)^(k-1) * k^(n-k),{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Jul 15 2014 *)
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a(n,m=1)=sum(k=0,n,binomial(n,k)*m*(n+m)^(k-1)*k^(n-k));
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/* Log(A(x)) = Sum_{n>=1} L(n)*x^n/n! where: */
L(n)=if(n<1,0,sum(k=1,n,binomial(n,k)*n^(k-1)*k^(n-k)));
A140049
E.g.f. A(x) satisfies: A( x*exp(-x*A(x)) ) = exp(x*A(x)).
Original entry on oeis.org
1, 1, 5, 55, 1005, 26601, 941863, 42372177, 2336926665, 153927536545, 11869936146891, 1055015092106889, 106731589524249517, 12163935655214359329, 1548324822731892094191, 218516875165035215308801, 33979477899236956531288977, 5790103152487972170694748097
Offset: 0
A(x) = 1 + x + 5*x^2/2! + 55*x^3/3! + 1005*x^4/4! + 26601*x^5/5! +...
Log(A(x)) = G(x) = e.g.f. of A140055:
Log(A(x)) = x + 4*x^2/2! + 42*x^3/3! + 764*x^4/4! + 20400*x^5/5! +...
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b:= proc(n, k) option remember; `if`(n=0, 1/k, add(k*j
*b(j-1, j)*b(n-j, k)*binomial(n-1, j-1), j=1..n))
end:
a:= n-> b(n, n+1):
seq(a(n), n=0..20); # Alois P. Heinz, Aug 21 2019
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m = 18; A[_] = 0;
Do[A[x_] = Exp[x A[x] A[x A[x]]] + O[x]^m // Normal, {m}];
CoefficientList[A[x], x] * Range[0, m-1]! (* Jean-François Alcover, Oct 03 2019 *)
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{a(n)=local(A=x);for(i=0,n,A=serreverse(x*exp(-A+x*O(x^n))));n!*polcoeff(A,n+1)}
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{a(n)=local(A=x);for(i=0,n,A=x*exp(subst(A,x,A+x*O(x^n))));n!*polcoeff(A,n+1)}
From Paul D. Hanna, Jul 09 2009: (Start)
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{a(n,m=1)=if(n==0,1,if(m==0,0^n,sum(k=0,n,binomial(n,k)*m*(n+m)^(k-1)*a(n-k,k))))}
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/* Log(A(x)) = x*A(x*A(x)) = Sum_{n>=1} L(n)*x^n/n! where: */
{L(n)=if(n<1,0,sum(k=1,n,binomial(n,k)*n^(k-1)*a(n-k,k)))} (End)
A384719
E.g.f. A(x) satisfies A(x) = exp( x * A(x*A(x))^2 ).
Original entry on oeis.org
1, 1, 5, 61, 1281, 39641, 1655713, 88312869, 5792082817, 454510418545, 41802078248001, 4434246169988669, 535583662477158529, 72887981688629021097, 11079094119653898282337, 1867050981690536859738901, 346619463962928284995333377, 70501622878003227432547203809
Offset: 0
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a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, (n-j+k)^(j-1)*binomial(n, j)*a(n-j, 2*j)));
A384720
E.g.f. A(x) satisfies A(x) = exp( x * A(x*A(x))^3 ).
Original entry on oeis.org
1, 1, 7, 118, 3385, 141556, 7918489, 561302470, 48589734337, 5001284972872, 599865865782481, 82534986682048066, 12863925185682542833, 2248009460254706256460, 436716594440553989797369, 93635975845903995553159126, 22021353830468757164023479169, 5650417076648052544704264390160
Offset: 0
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a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, (n-j+k)^(j-1)*binomial(n, j)*a(n-j, 3*j)));
A384739
E.g.f. A(x) satisfies A(x) = exp( x * A(x*A(x)^2) ).
Original entry on oeis.org
1, 1, 3, 28, 461, 11776, 421207, 19832128, 1179482201, 85990657024, 7513043962571, 772836266189824, 92270347493126629, 12636256749099114496, 1965364897138717976735, 344225592620170387849216, 67392512492360201909759153, 14653181755453024592646111232, 3518079370651785227796264294163
Offset: 0
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a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, (2*n-2*j+k)^(j-1)*binomial(n, j)*a(n-j, j)));
A384740
E.g.f. A(x) satisfies A(x) = exp( x * A(x*A(x)^3) ).
Original entry on oeis.org
1, 1, 3, 34, 665, 20556, 901417, 52455250, 3885229665, 355223077336, 39166024398641, 5113078496932374, 778733373110049601, 136679150176555902436, 27360426865918664532393, 6191378995818235673842546, 1571577905668087973855557313, 444441393534829346316950781744
Offset: 0
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a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, (3*n-3*j+k)^(j-1)*binomial(n, j)*a(n-j, j)));
A384757
E.g.f. A(x) satisfies A(x) = exp( -x * A(-x*A(x)) ).
Original entry on oeis.org
1, -1, -1, 14, 9, -1516, 4345, 507870, -4984063, -367545880, 7749976401, 471799390490, -18036953224367, -948817553760324, 60774529797257081, 2736041193224490494, -284790488755979731455, -10493764378757426300848, 1792499910367109444364961, 49177040508763120698604578
Offset: 0
E.g.f.: A(x) = 1 - x - x^2/2 + 14*x^3/6 + 9*x^4/24 - 1516*x^5/120 + ...
A(-x*A(x)) = 1 + x - 3*x^2/2 - 11*x^3/6 + 233*x^4/24 + 621*x^5/120 - ...
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a(n, k=1) = if(k==0, 0^n, (-1)^n*k*sum(j=0, n, (n-j+k)^(j-1)*binomial(n, j)*a(n-j, j)));
Showing 1-8 of 8 results.