A360349
G.f. A(x) = exp( Sum_{k>=1} A360348(k) * x^k/k ), where A360348(k) = [y^k*x^k/k] log( Sum_{m>=0} (1 + m*y + y^2)^m * x^m ) for k >= 1.
Original entry on oeis.org
1, 1, 5, 38, 391, 5077, 79535, 1458264, 30621237, 724555611, 19076629520, 553236991215, 17525729241605, 602215048797900, 22312035980459259, 886733059906749795, 37631474149766344476, 1698581174869953607957, 81257725943229600518977, 4106922637708383448243974
Offset: 0
G.f.: A(x) = 1 + x + 5*x^2 + 38*x^3 + 391*x^4 + 5077*x^5 + 79535*x^6 + 1458264*x^7 + 30621237*x^8 + 724555611*x^9 + ...
such that
log(A(x)) = x + 9*x^2/2 + 100*x^3/3 + 1381*x^4/4 + 22771*x^5/5 + 435138*x^6/6 + 9442049*x^7/7 + 229265109*x^8/8 + ... + A360348(n)*x^n/n + ...
where A360348(n) equals the coefficient of y^n*x^n/n in the logarithmic series:
log( Sum_{m>=0} (1 + m*y + y^2)^m * x^m ) = (y^2 + y + 1)*x + (y^4 + 6*y^3 + 9*y^2 + 6*y + 1)*x^2/2 + (y^6 + 15*y^5 + 63*y^4 + 100*y^3 + 63*y^2 + 15*y + 1)*x^3/3 + (y^8 + 28*y^7 + 242*y^6 + 872*y^5 + 1381*y^4 + 872*y^3 + 242*y^2 + 28*y + 1)*x^4/4 + (y^10 + 45*y^9 + 665*y^8 + 4430*y^7 + 14545*y^6 + 22771*y^5 + 14545*y^4 + 4430*y^3 + 665*y^2 + 45*y + 1)*x^5/5 + ...
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{A360348(n) = n * polcoeff( polcoeff( log( sum(m=0, n+1, (1 + m*y + y^2)^m *x^m ) +x*O(x^n) ), n, x), n, y)}
{a(n) = polcoeff( exp( sum(m=1,n, A360348(m)*x^m/m ) +x*O(x^n)),n)}
for(n=0,20,print1(a(n),", "))
A360238
a(n) = [y^n*x^n/n] log( Sum_{m>=0} (m + y)^(2*m) * x^m ) for n >= 1.
Original entry on oeis.org
2, 42, 1376, 60934, 3377252, 224036904, 17282039280, 1519096411230, 149867251224092, 16398595767212452, 1971137737765484444, 258215735255164847944, 36617351885600586385222, 5588967440618883091216208, 913592455995572681826313856, 159241707066923571547572653630
Offset: 1
L.g.f.: A(x) = 2*x + 42*x^2/2 + 1376*x^3/3 + 60934*x^4/4 + 3377252*x^5/5 + 224036904*x^6/6 + 17282039280*x^7/7 + 1519096411230*x^8/8 + ...
a(n) equals the coefficient of y^n*x^n/n in the logarithmic series:
log( Sum_{m>=0} (m + y)^(2*m) * x^m ) = (y^2 + 2*y + 1)*x + (y^4 + 12*y^3 + 42*y^2 + 60*y + 31)*x^2/2 + (y^6 + 30*y^5 + 297*y^4 + 1376*y^3 + 3348*y^2 + 4188*y + 2140)*x^3/3 + (y^8 + 56*y^7 + 1100*y^6 + 10792*y^5 + 60934*y^4 + 209464*y^3 + 436692*y^2 + 510952*y + 258779)*x^4/4 + (y^10 + 90*y^9 + 2945*y^8 + 49960*y^7 + 510160*y^6 + 3377252*y^5 + 14971780*y^4 + 44457000*y^3 + 85336175*y^2 + 96141170*y + 48446971)*x^5/5 + (y^12 + 132*y^11 + 6486*y^10 + 169236*y^9 + 2730921*y^8 + 29547696*y^7 + 224036904*y^6 + 1214958240*y^5 + 4717830978*y^4 + 12868488144*y^3 + 23497266672*y^2 + 25858665696*y + 12994749280)*x^6/6 + ...
Exponentiation yields the g.f. of A360239:
exp(A(x)) = 1 + 2*x + 23*x^2 + 502*x^3 + 16414*x^4 + 716936*x^5 + 39167817*x^6 + 2567058766*x^7 + 196159319943*x^8 + ... + A360239(n)*x^n + ...
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{a(n) = n * polcoeff( polcoeff( log( sum(m=0, n+1, (m + y)^(2*m) *x^m ) +x*O(x^n) ), n, x), n, y)}
for(n=0,20,print1(a(n),", "))
A360232
G.f. Sum_{n>=0} a(n)*x^n = Sum_{n>=0} (1 + n*x + x^2)^n * x^n.
Original entry on oeis.org
1, 1, 2, 6, 16, 51, 172, 626, 2409, 9791, 41671, 185224, 855865, 4100761, 20314349, 103827684, 546388333, 2955518901, 16407286272, 93350267922, 543674327227, 3237568471183, 19693508812475, 122249256779882, 773797772369256, 4990290667614087, 32766888950422831
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 16*x^4 + 51*x^5 + 172*x^6 + 626*x^7 + 2409*x^8 + 9791*x^9 + 41671*x^10 + 185224*x^11 + 855865*x^12 + ...
where
A(x) = 1 + (1 + x + x^2)*x + (1 + 2*x + x^2)^2*x^2 + (1 + 3*x + x^2)^3*x^3 + (1 + 4*x + x^2)^4*x^4 + ... + (1 + n*x + x^2)^n*x^n + ...
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nmax = 30; CoefficientList[Series[Sum[(1 + k*x + x^2)^k * x^k, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 13 2023 *)
Flatten[{1, Table[Sum[Sum[Binomial[k,j] * Binomial[j,n-k-j] * k^(2*j - n + k), {j, 0, k}], {k, 1, n}], {n, 1, 30}]}] (* Vaclav Kotesovec, Feb 14 2023 *)
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{a(n) = polcoeff( sum(m=0,n, (1 + m*x + x^2)^m * x^m +x*O(x^n)),n)}
for(n=0,30,print1(a(n),", "))
Showing 1-3 of 3 results.
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