Search: a318633 -id:a318633
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A266485
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E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N + 2*n)^(2*n) * (x/N)^n/n! ]^(1/N).
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+10
10
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1, 1, 9, 193, 6929, 356001, 24004825, 2012327521, 202156421409, 23701550853313, 3179302948594601, 480443117415138945, 80788534008942735409, 14965275494082095616097, 3028424508967743713615481, 664790043100841638943719201, 157352199248412053285546165825, 39950540529265571984889165180801
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OFFSET
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0,3
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COMMENTS
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Compare to: Limit_{N->oo} [ Sum_{n>=0} (N + n)^n * x^n/n! ]^(1/N) = Sum_{n>=0} (n+1)^(n-1) * x^n/n!.
exp(x) = lim_{N->oo} [ Sum_{n>=0} (N^2 + n)^n * (x/N)^n/n! ]^(1/N).
W(x) = lim_{N->oo} [ Sum_{n>=0} (N^2 + N*n)^n * (x/N)^n/n! ]^(1/N), where W(x) = LambertW(-x)/(-x).
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LINKS
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FORMULA
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E.g.f. A(x) = Sum_{n>=0} a(n) * x^n/n! may be defined by the following (Paul D. Hanna, Jan 20 2023):
(1) A(x) = lim_{N->oo} [ Sum_{n>=0} (N + 2*n)^(2*n) * (x/N)^n/n! ]^(1/N).
(2) A(x) = exp( Sum_{n>=0} A359926(n)*x^n/n! ), where A359926(n) = (1/4) * [x^n*y^(n+1)/n!] log( Sum_{n>=0} (n + 2*y)^(2*n) *x^n/n! ).
a(n) ~ c * d^n * n^(n-2), where d = 4*(1 + sqrt(2)) * exp(1 - sqrt(2)) = 6.3818267815342167443903123351857161682971406064645602440616... and c = sqrt(1 - 1/sqrt(2))/2 * exp(3/2 - sqrt(2)) = 0.294836494691148677397464568534316405253091784834436235... - Vaclav Kotesovec, Jan 21 2023, updated Mar 17 2024
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EXAMPLE
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E.g.f.: A(x) = 1 + x + 9*x^2/2! + 193*x^3/3! + 6929*x^4/4! + 356001*x^5/5! + 24004825*x^6/6! + 2012327521*x^7/7! + 202156421409*x^8/8! + 23701550853313*x^9/9! + 3179302948594601*x^10/10! +...
where A(x) equals the limit, as N -> oo, of the series
[1 + (N+2)^2*(x/N) + (N+4)^4*(x/N)^2/2! + (N+6)^6*(x/N)^3/3! + (N+8)^8*(x/N)^4/4! + (N+10)^10*(x/N)^5/5! + (N+12)^12*(x/N)^6/6! +...]^(1/N).
The logarithm of the g.f. A(x) begins (Paul D. Hanna, Jan 20 2023):
(a) log(A(x)) = x + 8*x^2/2! + 168*x^3/3! + 6016*x^4/4! + 309760*x^5/5! + 20957184*x^6/6! + 1762991104*x^7/7! + 177690607616*x^8/8! + ... + A359926(n)*x^n/n! + ...
where A359926(n) = [x^n*y^(n+1)/n!] (1/4) * log( Sum_{n>=0} (n + 2*y)^(2*n) * x^n/n! );
that is, the coefficient of x^n/n! in the logarithm of e.g.f A(x) equals the coefficient of y^(n+1)*x^n/n! in the series given by
(b) (1/4) * log( Sum_{n>=0} (n + 2*y)^(2*n) * x^n/n! ) = (y^2 + y + 1/4)*x + (8*y^3 + 18*y^2 + 14*y + 15/4)*x^2/2! + (168*y^4 + 632*y^3 + 933*y^2 + 639*y + 683/4)*x^3/3! + (6016*y^5 + 33088*y^4 + 76480*y^3 + 92680*y^2 + 58720*y + 31019/2)*x^4/4! + ...
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PROG
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(PARI) /* Informal listing of terms 0..30 */
\p300
P(n) = sum(k=0, 32, (n+2*k)^(2*k) * x^k/k! +O(x^31))
Vec( round( serlaplace( subst(P(10^100)^(1/10^100), x, x/10^100) )*1.) )
(PARI) /* Using formula for the logarithm of g.f. A(x) Paul D. Hanna, Jan 20 2023 */
{L(n) = (1/4) * n! * polcoeff( polcoeff( log( sum(m=0, n+1, (m + 2*y)^(2*m) *x^m/m! ) +x*O(x^n) ), n, x), n+1, y)}
{a(n) = n! * polcoeff( exp( sum(m=1, n+1, L(m)*x^m/m! ) +x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))
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CROSSREFS
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Cf. A266481, A266482, A266483, A266484, A266486, A266487, A359926, A359927, A319147, A318633, A319834.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A318634
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a(n) = coefficient of x^(2*n-1)*y^(2*n)/(2*n-1)! in Log( Sum_{n>=0} (n^2 + y^2)^n * x^n/n! ), for n>=1.
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+10
6
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1, 6, 480, 122640, 66044160, 61482516480, 88135315107840, 180378921026304000, 499734635092800307200, 1801642618822079338905600, 8199046303785011864744755200, 45976521975711536997953490124800, 311502479360401852390993821696000000, 2508845886467091418046335123571343360000, 23693183471722887844366765687378500648960000
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OFFSET
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1,2
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COMMENTS
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E.g.f. A(x) = Sum_{n>=1} a(n)*x^(2*n-1)/(2*n-1)! equals the logarithm of the e.g.f. of A318633.
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LINKS
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FORMULA
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a(n) ~ 5^(-1/4) * 2^(3*n - 3/2) * (1 + sqrt(5))^(n - 3/2) * exp((1 - sqrt(5))*n + (sqrt(5) - 3)/2) * n^(2*n-3). - Vaclav Kotesovec, Mar 20 2024
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EXAMPLE
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E.g.f.: A(x) = x + 6*x^3/3! + 480*x^5/5! + 122640*x^7/7! + 66044160*x^9/9! + 61482516480*x^11/11! + 88135315107840*x^13/13! + 180378921026304000*x^15/15! + ...
The e.g.f. A(x) may also be written using somewhat reduced coefficients
A(x) = x + x^3 + 8*x^5/2! + 146*x^7/3! + 4368*x^9/4! + 184832*x^11/5! + 10190656*x^13/6! + 695211120*x^15/7! + 56648897024*x^17/8! + 5374487515904*x^19/9! + ... + a(n)*(n-1)!/(2*n-1)! * x^(2*n-1)/(n-1)! + ...
Exponentiation yields the e.g.f. of A318633:
exp(A(x)) = 1 + x + x^2/2! + 7*x^3/3! + 25*x^4/4! + 541*x^5/5! + 3361*x^6/6! + 135451*x^7/7! + 1179697*x^8/8! + 72062425*x^9/9! +...+ A318633(n)*x^n/n! + ...
which equals
Limit_{N->oo} [ Sum_{n>=0} (N^2 + n^2)^n * (x/N)^n/n! ]^(1/N).
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PROG
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(PARI) {a(n) = (2*n-1)! * polcoeff( polcoeff( log( sum(m=0, 2*n, (m^2 + y^2)^m *x^m/m! ) +x*O(x^(2*n)) ), 2*n-1, x), 2*n, y)}
for(n=1, 20, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A319147
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E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N^2 + N*n + n^2)^n * (x/N)^n/n! ]^(1/N).
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+10
6
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1, 1, 3, 22, 269, 4776, 111967, 3280264, 115550073, 4762181440, 224954474651, 11987717900544, 711604917300037, 46572971758429312, 3332107859592406455, 258748811312125854976, 21674785904235983431793, 1948303837796264786497536, 187062919027712092164076723, 19107058023481400501276569600
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OFFSET
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0,3
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COMMENTS
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Compare to:
(C1) exp(x) = Limit_{N->oo} [ Sum_{n>=0} (N^2 + n)^n * (x/N)^n/n! ]^(1/N).
(C2) W(x) = Limit_{N->oo} [ Sum_{n>=0} (N^2 + N*n)^n * (x/N)^n/n! ]^(1/N), where W(x) = LambertW(-x)/(-x).
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LINKS
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FORMULA
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E.g.f. exp( Sum_{n>=0} L(n)*x^n/n! ), where L(n) = [x^n*y^(n+1)/n!] log( Sum_{n>=0} (n^2 + n*y + y^2)^n *x^n/n! ).
a(n) ~ c * d^n * n! / n^(5/2), where d = 6.16018341007619464488... and c = 0.240315927519139896... - Vaclav Kotesovec, Mar 19 2022
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EXAMPLE
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E.g.f.: A(x) = 1 + x + 3*x^2/2! + 22*x^3/3! + 269*x^4/4! + 4776*x^5/5! + 111967*x^6/6! + 3280264*x^7/7! + 115550073*x^8/8! + 4762181440*x^9/9! + 224954474651*x^10/10! + ...
where A(x) equals the limit, as N -> oo, of the series
[1 + (N^2+N+1)*(x/N) + (N^2+2*N+2^2)^2*(x/N)^2/2! + (N^2+3*N+3^2)^3*(x/N)^3/3! + (N^2+4*N+4^2)^4*(x/N)^4/4! + (N^2+5*N+5^2)^5*(x/N)^5/5! + (N^2+6*N+6^2)^6*(x/N)^6/6! +...]^(1/N).
RELATED SERIES.
(a) The logarithm of the g.f. A(x) begins:
log(A(x)) = x + 2*x^2/2! + 15*x^3/3! + 184*x^4/4! + 3325*x^5/5! + 79056*x^6/6! + 2345539*x^7/7! + 83505920*x^8/8! + 3472829721*x^9/9! + ... + A319834(n)*x^n/n! + ...
where A319834(n) = [x^n*y^(n+1)/n!] log( Sum_{n>=0} (n^2 + n*y + y^2)^n * x^n/n! );
that is, the coefficients in the logarithm of e.g.f A(x) equals the coefficients of y^(n+1)*x^n/n! in the series given by
(b) log( Sum_{n>=0} (n^2 + n*y + y^2)^n * x^n/n! ) = (y^2 + y + 1)*x + (2*y^3 + 9*y^2 + 14*y + 15)*x^2/2! + (15*y^4 + 107*y^3 + 366*y^2 + 639*y + 683)*x^3/3! + (184*y^5 + 2038*y^4 + 10432*y^3 + 32308*y^2 + 58720*y + 62038)*x^4/4! + (3325*y^6 + 50469*y^5 + 367155*y^4 + 1636590*y^3 + 4833195*y^2 + 8940045*y + 9342629)*x^5/5! + (79056*y^7 + 1565256*y^6 + 15015936*y^5 + 90978240*y^4 + 376955520*y^3 + 1085556216*y^2 + 2027376336*y + 2100483216)*x^6/6! + ...
(c) The following limit exists
G(x) = Limit_{N->oo} [ Sum_{n>=0} (N^2 + N*n + n^2)^n * (x/N)^n/n! ] / A(x)^N
where
G(x) = 1 + x + 10*x^2/2! + 135*x^3/3! + 2764*x^4/4! + 72665*x^5/5! + 2362896*x^6/6! + 91282975*x^7/7! + 4088186320*x^8/8! + 208223576721*x^9/9! + ...
the logarithm of which begins
log(G(x)) = x + 9*x^2/2! + 107*x^3/3! + 2038*x^4/4! + 50469*x^5/5! + 1565256*x^6/6! + 58095463*x^7/7! + 2513768496*x^8/8! + ... + D(n)*x^n/n! + ...
where D(n) = [x^n*y^n/n!] log( Sum_{n>=0} (n^2 + n*y + y^2)^n * x^n/n! ).
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PROG
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(PARI) {L(n) = n! * polcoeff( polcoeff( log( sum(m=0, n+1, (m^2 + m*y + y^2)^m *x^m/m! ) +x*O(x^n) ), n, x), n+1, y)}
{a(n) = n! * polcoeff( exp( sum(m=1, n+1, L(m)*x^m/m! ) +x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* Informal listing of terms 0..30 */
\p100
P(n) = sum(k=0, 31, (n^2 + n*k + k^2)^k * x^k/k! +O(x^31))
Vec(round( serlaplace( subst(P(10^100)^(1/10^100), x, x/10^100) )*1.) )
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A359927
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E.g.f.: lim_{N->oo} [ Sum_{n>=0} (N^2 + 3*N*n + 2*n^2)^n * (x/N)^n/n! ]^(1/N).
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+10
4
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1, 1, 7, 112, 2965, 111856, 5528419, 339433984, 24965493865, 2142654088960, 210377086601311, 23269631260880896, 2864038963868253373, 388330717110688399360, 57521524729462484086075, 9242821569458332441378816, 1601434996324769244061560529
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OFFSET
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0,3
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COMMENTS
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Related limits:
(C1) exp(x) = lim_{N->oo} [ Sum_{n>=0} (N^2 + n)^n * (x/N)^n/n! ]^(1/N).
(C2) W(x) = lim_{N->oo} [ Sum_{n>=0} (N^2 + N*n)^n * (x/N)^n/n! ]^(1/N), where W(x) = LambertW(-x)/(-x).
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LINKS
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FORMULA
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E.g.f. A(x) = Sum_{n>=0} a(n) * x^n/n! may be defined by the following.
(1) A(x) = lim_{N->oo} [ Sum_{n>=0} (N + n)^n*(N + 2*n)^n * (x/N)^n/n! ]^(1/N).
(2) A(x) = exp( Sum_{n>=0} A359928(n)*x^n/n! ), where A359928(n) = (1/2) * [x^n*y^(n+1)/n!] log( Sum_{n>=0} (n + y)^n*(n + 2*y)^n *x^n/n! ).
a(n) ~ c * n! * d^n / n^(5/2), where d = 12.7029497597456784744445675253711147535742245945208995646... and c = 0.17380315134029681101563539591890111670852050181568... - Vaclav Kotesovec, Mar 14 2023
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EXAMPLE
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E.g.f.: A(x) = 1 + x + 7*x^2/2! + 112*x^3/3! + 2965*x^4/4! + 111856*x^5/5! + 5528419*x^6/6! + 339433984*x^7/7! + 24965493865*x^8/8! + 2142654088960*x^9/9! + 210377086601311*x^10/10! + 23269631260880896*x^11/11! + 2864038963868253373*x^12/12! + ...
where A(x) equals the limit, as N -> oo, of the series
[1 + (N^2+3*N+2)*(x/N) + (N^2+3*2*N+2*2^2)^2*(x/N)^2/2! + (N^2+3*3*N+2*3^2)^3*(x/N)^3/3! + (N^2+3*4*N+2*4^2)^4*(x/N)^4/4! + (N^2+3*5*N+2*5^2)^5*(x/N)^5/5! + (N^2+3*6*N+2*6^2)^6*(x/N)^6/6! + ...]^(1/N).
RELATED SERIES.
The logarithm of the g.f. A(x) begins:
(a) log(A(x)) = x + 6*x^2/2! + 93*x^3/3! + 2448*x^4/4! + 92505*x^5/5! + 4589568*x^6/6! + 283008621*x^7/7! + 20903023872*x^8/8! + ... + A359928(n)*x^n/n! + ...
where A359928(n) = [x^n*y^(n+1)/n!] (1/2) * log( Sum_{n>=0} (n^2 + 3*n*y + 2*y^2)^n * x^n/n! );
that is, the coefficient of x^n/n! in the logarithm of e.g.f A(x) equals the coefficient of y^(n+1)*x^n/n!, n >= 1, in the series given by
(b) (1/2) * log( Sum_{n>=0} (n^2 + 3*n*y + 2*y^2)^n * x^n/n! ) = x*(y^2 + 3/2*y + 1/2) + x^2/2!*(6*y^3 + 39/2*y^2 + 21*y + 15/2) + x^3/3!*(93*y^4 + 999/2*y^3 + 2055/2*y^2 + 1917/2*y + 683/2) + x^4/4!*(2448*y^5 + 19119*y^4 + 61704*y^3 + 102742*y^2 + 88080*y + 31019) + x^5/5!*(92505*y^6 + 1948347/2*y^5 + 8887325/2*y^4 + 11224575*y^3 + 16525750*y^2 + 26820135/2*y + 9342629/2) + x^6/6!*(4589568*y^7 + 61994772*y^6 + 374546664*y^5 + 1310466240*y^4 + 2862046080*y^3 + 3891543876*y^2 + 3041064504*y + 1050241608) + ...
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PROG
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(PARI) /* Using formula for the logarithm of g.f. A(x) */
{L(n) = (1/2) * n! * polcoeff( polcoeff( log( sum(m=0, n+1, (m + y)^m*(m + 2*y)^m *x^m/m! ) +x*O(x^n) ), n, x), n+1, y)}
{a(n) = n! * polcoeff( exp( sum(m=1, n+1, L(m)*x^m/m! ) +x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* Using limit formula */
\p100
P(n) = sum(k=0, 31, ((n + k)*(n + 2*k))^k * x^k/k! +O(x^31))
Vec(round( serlaplace( subst(P(10^100)^(1/10^100), x, x/10^100) )*1.) )
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A359917
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E.g.f.: lim_{N->oo} [ Sum_{n>=0} (N^2 + N*n + 2*n^2)^n * (x/N)^n/n! ]^(1/N).
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+10
2
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1, 1, 3, 28, 413, 9216, 268327, 9831424, 432251577, 22259307520, 1313366140331, 87431498993664, 6482838033725077, 529958491541291008, 47356678577690489295, 4592761099982656823296, 480465410003489098874993, 53933291626260492656050176, 6466413087139041540884403667
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OFFSET
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0,3
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COMMENTS
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Related limits:
(C1) exp(x) = lim_{N->oo} [ Sum_{n>=0} (N^2 + n)^n * (x/N)^n/n! ]^(1/N).
(C2) W(x) = lim_{N->oo} [ Sum_{n>=0} (N^2 + N*n)^n * (x/N)^n/n! ]^(1/N), where W(x) = LambertW(-x)/(-x).
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LINKS
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FORMULA
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E.g.f. A(x) = Sum_{n>=0} a(n) * x^n/n! may be defined by the following.
(1) A(x) = lim_{N->oo} [ Sum_{n>=0} (N^2 + N*n + 2*n^2)^n * (x/N)^n/n! ]^(1/N).
(2) A(x) = exp( Sum_{n>=0} A359918(n)*x^n/n! ), where A359918(n) = (1/2) * [x^n*y^(n+1)/n!] log( Sum_{n>=0} (n^2 + n*y + 2*y^2)^n * x^n/n! ).
a(n) ~ c * d^n * n! / n^(5/2), where d = 7.68892218919697462312... and c = 0.155267010681833... - Vaclav Kotesovec, Mar 21 2024
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EXAMPLE
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E.g.f.: A(x) = 1 + x + 3*x^2/2! + 28*x^3/3! + 413*x^4/4! + 9216*x^5/5! + 268327*x^6/6! + 9831424*x^7/7! + 432251577*x^8/8! + 22259307520*x^9/9! + 1313366140331*x^10/10! + ...
where A(x) equals the limit, as N -> oo, of the series
[1 + (N^2+N+2)*(x/N) + (N^2+2*N+2*2^2)^2*(x/N)^2/2! + (N^2+3*N+2*3^2)^3*(x/N)^3/3! + (N^2+4*N+2*4^2)^4*(x/N)^4/4! + (N^2+5*N+2*5^2)^5*(x/N)^5/5! + (N^2+6*N+2*6^2)^6*(x/N)^6/6! + ...]^(1/N).
RELATED SERIES.
The logarithm of the g.f. A(x) begins:
(a) log(A(x)) = x + 2*x^2/2! + 21*x^3/3! + 304*x^4/4! + 6985*x^5/5! + 205056*x^6/6! + 7607509*x^7/7! + ... + A359918(n)*x^n/n! + ...
where A359918(n) = [x^n*y^(n+1)/n!] (1/2) * log( Sum_{n>=0} (n^2 + n*y + 2*y^2)^n * x^n/n! );
that is, the coefficients in the logarithm of e.g.f A(x) equals the coefficients of y^(n+1)*x^n/n! in the series given by
(b) (1/2) * log( Sum_{n>=0} (n^2 + n*y + 2*y^2)^n * x^n/n! ) = (y^2 + 1/2*y + 1/2)*x + (2*y^3 + 15/2*y^2 + 7*y + 15/2)*x^2/2! + (21*y^4 + 197/2*y^3 + 543/2*y^2 + 639/2*y + 683/2)*x^3/3! + (304*y^5 + 2495*y^4 + 8984*y^3 + 22246*y^2 + 29360*y + 31019)*x^4/4! + (6985*y^6 + 150489/2*y^5 + 817005/2*y^4 + 1335885*y^3 + 3162830*y^2 + 8940045/2*y + 9342629/2)*x^5/5! + (205056*y^7 + 2946228*y^6 + 20587128*y^5 + 94146240*y^4 + 294518400*y^3 + 684700836*y^2 + 1013688168*y + 1050241608)*x^6/6! + ...
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PROG
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(PARI) /* Using formula for the logarithm of g.f. A(x) */
{L(n) = (1/2) * n! * polcoeff( polcoeff( log( sum(m=0, n+1, (m^2 + m*y + 2*y^2)^m *x^m/m! ) +x*O(x^n) ), n, x), n+1, y)}
{a(n) = n! * polcoeff( exp( sum(m=1, n+1, L(m)*x^m/m! ) +x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* Using limit formula */
\p100
P(n) = sum(k=0, 31, (n^2 + n*k + 2*k^2)^k * x^k/k! +O(x^31))
Vec(round( serlaplace( subst(P(10^100)^(1/10^100), x, x/10^100) )*1.) )
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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