A007841
Number of factorizations of permutations of n letters into cycles in nondecreasing length order.
Original entry on oeis.org
1, 1, 3, 11, 56, 324, 2324, 18332, 167544, 1674264, 18615432, 223686792, 2937715296, 41233157952, 623159583552, 10008728738304, 171213653641344, 3092653420877952, 59086024678203264, 1185657912197967744, 25015435198774723584, 552130504313534175744
Offset: 0
- Alois P. Heinz, Table of n, a(n) for n = 0..450
- Vaclav Kotesovec, Graph - The asymptotic ratio
- A. Knopfmacher, J. N. Ridley, Reciprocal sums over partitions and compositions, SIAM J. Discrete Math. 6 (1993), no. 3, 388-399.
- D. H. Lehmer, On reciprocally weighted partitions, Acta Arithmetica XXI (1972), 379-388.
-
p := product(1/(1-x^m/m), m=1..100):
s := series(p,x,100):
for i from 0 to 100 do printf(`%.0f,`,i!*coeff(s,x,i)) od:
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
(i-1)!^j*b(n-i*j, i-1)*multinomial(n, n-i*j, i$j), j=0..n/i)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..30); # Alois P. Heinz, Jul 21 2014
-
nmax = 25; CoefficientList[Series[1/Product[(1 - x^k/k), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jul 24 2019 *)
nmax = 25; CoefficientList[Series[Exp[Sum[PolyLog[j, x^j]/j, {j, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jul 24 2019 *)
-
R(n,m):=if n=0 then 1 else if nVladimir Kruchinin, Sep 09 2014 */
-
N=66; q='q+O('q^N);
f=1/prod(n=1,N, 1-1/n*q^n );
egf=serlaplace(f);
Vec(egf)
/* Joerg Arndt, Oct 06 2012 */
A249474
E.g.f.: P(x)/exp(1) + Q(x), where P(x) = 1/Product_{n>=1} (1 - x^n/n) and Q(x) = Sum_{n>=1} -(-1)^n / Product_{k=1..n} (k - x^k).
Original entry on oeis.org
1, 1, 2, 7, 30, 169, 1128, 8700, 76494, 753139, 8182188, 97131376, 1256860330, 17470791933, 261284377168, 4164406202270, 70677340199670, 1268718107324255, 24091289738163140, 480954355282406340, 10097484764045220626, 221918808641500960217, 5103937368681669463800
Offset: 0
E.g.f.: A(x) = 1 + x + 2*x^2/2! + 7*x^3/3! + 30*x^4/4! + 169*x^5/5! +...
such that A(x) = exp(-1)*P(x) + Q(x), where
P(x) = 1/Product_{n>=1} (1 - x^n/n) = Sum_{n>=0} A007841(n)*x^n/n!, and
Q(x) = Sum_{n>=1} -(-1)^n / Product_{k=1..n} (k - x^k).
More explicitly,
P(x) = 1/((1-x)*(1-x^2/2)*(1-x^3/3)*(1-x^4/4)*(1-x^5/5)*...);
Q(x) = 1/(1-x) - 1/((1-x)*(2-x^2)) + 1/((1-x)*(2-x^2)*(3-x^3)) - 1/((1-x)*(2-x^2)*(3-x^3)*(4-x^4)) + 1/((1-x)*(2-x^2)*(3-x^3)*(4-x^4)*(5-x^5)) +-...
We can illustrate the initial terms a(n) in the following manner.
The coefficients in Q(x) = Sum_{n>=0} q(n)*x^n/n! begin:
q(0) = 0.632120558828557678...
q(1) = 0.632120558828557678...
q(2) = 0.896361676485673035...
q(3) = 2.953326147114134462...
q(4) = 9.398751294399229990...
q(5) = 49.80706106045268780...
q(6) = 273.0481787175680446...
q(7) = 1956.034084445119360...
q(8) = 14858.00690837186767...
q(9) = 137211.6953065362928...
and the coefficients in P(x) = 1/Product_{n>=1} (1 - x^n/n) begin:
A007841 = [1, 1, 3, 11, 56, 324, 2324, 18332, 167544, ...];
from which we can generate this sequence like so:
a(0) = exp(-1)*1 + q(0) = 1;
a(1) = exp(-1)*1 + q(1) = 1;
a(2) = exp(-1)*3 + q(2) = 2;
a(3) = exp(-1)*11 + q(3) = 7;
a(4) = exp(-1)*56 + q(4) = 30;
a(5) = exp(-1)*324 + q(5) = 169;
a(6) = exp(-1)*2324 + q(6) = 1128;
a(7) = exp(-1)*18332 + q(7) = 8700;
a(8) = exp(-1)*167544 + q(8) = 76494; ...
-
\p100 \\ set precision
{P=Vec(serlaplace(prod(k=1, 31, 1/(1-x^k/k +O(x^31))))); } \\ A007841
{Q=Vec(serlaplace(sum(n=1, 201, -(-1)^n * prod(k=1, n, 1./(k-x^k +O(x^31)))))); }
for(n=0, 30, print1(round(exp(-1)*P[n+1]+Q[n+1]), ", "))
A249475
E.g.f.: exp(2)*P(x) - Q(x), where P(x) = 1/Product_{n>=1} (1 - x^n/n) and Q(x) = Sum_{n>=1} 2^n/Product_{k=1..n} (k - x^k).
Original entry on oeis.org
1, 1, 5, 25, 156, 1048, 8400, 72384, 710184, 7519240, 87797880, 1098513880, 14945280640, 216079283040, 3352657547680, 55071779464352, 961293645943680, 17669716422651776, 342988501737128576, 6978772157389361280, 149123855108936024576, 3328674238745847019520
Offset: 0
E.g.f.: A(x) = 1 + x + 5*x^2/2! + 25*x^3/3! + 156*x^4/4! + 1048*x^5/5! +...
such that A(x) = exp(2)*P(x) - Q(x), where
P(x) = 1/Product_{n>=1} (1 - x^n/n) = Sum_{n>=0} A007841(n)*x^n/n!, and
Q(x) = Sum_{n>=1} 2^n / Product_{k=1..n} (k - x^k).
More explicitly,
P(x) = 1/((1-x)*(1-x^2/2)*(1-x^3/3)*(1-x^4/4)*(1-x^5/5)*...);
Q(x) = 2/(1-x) + 2^2/((1-x)*(2-x^2)) + 2^3/((1-x)*(2-x^2)*(3-x^3)) + 2^4/((1-x)*(2-x^2)*(3-x^3)*(4-x^4)) + 2^5/((1-x)*(2-x^2)*(3-x^3)*(4-x^4)*(5-x^5)) +...
We can illustrate the initial terms a(n) in the following manner.
The coefficients in Q(x) = Sum_{n>=0} q(n)*x^n/n! begin:
q(0) = 6.3890560989306502272...
q(1) = 6.3890560989306502272...
q(2) = 17.167168296791950681...
q(3) = 56.279617088237152499...
q(4) = 257.78714154011641272...
q(5) = 1346.0541760535306736...
q(6) = 8772.1663739148311280...
q(7) = 63072.176405596679965...
q(8) = 527808.01503923686167...
q(9) = 4851990.6204200261720...
and the coefficients in P(x) = 1/Product_{n>=1} (1 - x^n/n) begin:
A007841 = [1, 1, 3, 11, 56, 324, 2324, 18332, 167544, ...];
from which we can generate this sequence like so:
a(0) = exp(2)*1 - q(0) = 1;
a(1) = exp(2)*1 - q(1) = 1;
a(2) = exp(2)*3 - q(2) = 5;
a(3) = exp(2)*11 - q(3) = 25;
a(4) = exp(2)*56 - q(4) = 156;
a(5) = exp(2)*324 - q(5) = 1048;
a(6) = exp(2)*2324 - q(6) = 8400;
a(7) = exp(2)*18332 - q(7) = 72384;
a(8) = exp(2)*167544 - q(8) = 710184; ...
-
\p100 \\ set precision
{P=Vec(serlaplace(prod(k=1, 31, 1/(1-x^k/k +O(x^31))))); } \\ A007841
{Q=Vec(serlaplace(sum(n=1, 201, 2^n * prod(k=1, n, 1./(k-x^k +O(x^31)))))); }
for(n=0, 30, print1(round(exp(2)*P[n+1]-Q[n+1]), ", "))
A249480
E.g.f.: A(x,y) = exp(y)*P(x) - Q(x,y), where P(x) = 1/Product_{n>=1} (1 - x^n/n) and Q(x,y) = Sum_{n>=1} y^n / Product_{k=1..n} (k - x^k).
Original entry on oeis.org
1, 1, 0, 3, 1, 0, 11, 5, 1, 0, 56, 32, 7, 1, 0, 324, 204, 57, 9, 1, 0, 2324, 1604, 487, 89, 11, 1, 0, 18332, 13292, 4441, 897, 128, 13, 1, 0, 167544, 127224, 44712, 9864, 1486, 174, 15, 1, 0, 1674264, 1311384, 485592, 111744, 18486, 2286, 227, 17, 1, 0, 18615432, 14986632, 5735616, 1393872, 240318, 31734, 3329, 287, 19, 1, 0
Offset: 0
Triangle begins:
1;
1, 0;
3, 1, 0;
11, 5, 1, 0;
56, 32, 7, 1, 0;
324, 204, 57, 9, 1, 0;
2324, 1604, 487, 89, 11, 1, 0;
18332, 13292, 4441, 897, 128, 13, 1, 0;
167544, 127224, 44712, 9864, 1486, 174, 15, 1, 0;
1674264, 1311384, 485592, 111744, 18486, 2286, 227, 17, 1, 0;
18615432, 14986632, 5735616, 1393872, 240318, 31734, 3329, 287, 19, 1, 0;
223686792, 183769992, 72550296, 18223632, 3296958, 455742, 51009, 4647, 354, 21, 1, 0;
2937715296, 2458713696, 993598248, 257587416, 48076704, 6958656, 801880, 77896, 6272, 428, 23, 1, 0;
41233157952, 35006137152, 14438206776, 3835359192, 738870048, 110022696, 13300084, 1330300, 114164, 8236, 509, 25, 1, 0; ...
GENERATING FUNCTION.
G.f.: A(x,y) = 1 + (1)*x + (3 + y)*x^2/2! + (11 + 5*y + y^2)*x^3/3! +
(56 + 32*y + 7*y^2 + y^3)*x^4/4! +
(324 + 204*y + 57*y^2 + 9*y^3 + y^4)*x^5/5! +
(2324 + 1604*y + 487*y^2 + 89*y^3 + 11*y^4 + y^5)*x^6/6! +...
such that
A(x,y) = exp(y)*P(x) - Q(x,y)
where
P(x) = 1/Product_{n>=1} (1 - x^n/n) = Sum_{n>=0} A007841(n)*x^n/n!, and
Q(x,y) = Sum_{n>=1} y^n / Product_{k=1..n} (k - x^k).
More explicitly,
P(x) = 1/((1-x)*(1-x^2/2)*(1-x^3/3)*(1-x^4/4)*(1-x^5/5)*...)
Q(x,y) = y/(1-x) + y^2/((1-x)*(2-x^2)) + y^3/((1-x)*(2-x^2)*(3-x^3)) + y^4/((1-x)*(2-x^2)*(3-x^3)*(4-x^4)) + y^5/((1-x)*(2-x^2)*(3-x^3)*(4-x^4)*(5-x^5)) +...
Column zero of this triangle forms the e.g.f. of A007841:
P(x) = 1 + x + 3*x^2/2! + 11*x^3/3! + 56*x^4/4! + 324*x^5/5! + 2324*x^6/6! + 18332*x^7/7! + 167544*x^8/8! +...
-
{T(n,k)=local(A=1, P=((prod(j=1, n+1, 1/(1 - x^j/j +x^2*O(x^n))))),
Q=((sum(m=1, n+1, y^m * prod(j=1, m, 1/(j - x^j +x^2*O(x^n)))))) );
A=exp(y)*P - Q; n!*polcoeff(polcoeff(A,n,x),k,y)}
for(n=0,12,for(k=0,n,print1(T(n,k),", "));print(""))
A249476
E.g.f.: exp(3)*P(x) - Q(x), where P(x) = 1/Product_{n>=1} (1 - x^n/n) and Q(x) = Sum_{n>=1} 3^n/Product_{k=1..n} (k - x^k).
Original entry on oeis.org
1, 1, 6, 35, 242, 1773, 15056, 136652, 1393722, 15257919, 183206388, 2347929936, 32602306542, 479885400177, 7563888117504, 125952344438838, 2225653414414386, 41351620513521627, 810520833521436732, 16633643598838880244, 358221783030360367014, 8051927483267030640573
Offset: 0
E.g.f.: A(x) = 1 + x + 6*x^2/2! + 35*x^3/3! + 242*x^4/4! + 1773*x^5/5! +...
such that A(x) = exp(3)*P(x) - Q(x), where
P(x) = 1/Product_{n>=1} (1 - x^n/n) = Sum_{n>=0} A007841(n)*x^n/n!, and
Q(x) = Sum_{n>=1} 3^n / Product_{k=1..n} (k - x^k).
More explicitly,
P(x) = 1/((1-x)*(1-x^2/2)*(1-x^3/3)*(1-x^4/4)*(1-x^5/5)*...);
Q(x) = 3/(1-x) + 3^2/((1-x)*(2-x^2)) + 3^3/((1-x)*(2-x^2)*(3-x^3)) + 3^4/((1-x)*(2-x^2)*(3-x^3)*(4-x^4)) + 3^5/((1-x)*(2-x^2)*(3-x^3)*(4-x^4)*(5-x^5)) +...
We can illustrate the initial terms a(n) in the following manner.
The coefficients in Q(x) = Sum_{n>=0} q(n)*x^n/n! begin:
q(0) = 19.085536923187667740...
q(1) = 19.085536923187667740...
q(2) = 54.256610769563003222...
q(3) = 185.94090615506434515...
q(4) = 882.79006769850939349...
q(5) = 4734.7139631128043480...
q(6) = 31622.787809488139829...
q(7) = 231556.06287587632502...
q(8) = 1971489.1982585546039...
q(9) = 18370572.391163877342...
and the coefficients in P(x) = 1/Product_{n>=1} (1 - x^n/n) begin:
A007841 = [1, 1, 3, 11, 56, 324, 2324, 18332, 167544, ...];
from which we can generate this sequence like so:
a(0) = exp(3)*1 - q(0) = 1;
a(1) = exp(3)*1 - q(1) = 1;
a(2) = exp(3)*3 - q(2) = 6;
a(3) = exp(3)*11 - q(3) = 35;
a(4) = exp(3)*56 - q(4) = 242;
a(5) = exp(3)*324 - q(5) = 1773;
a(6) = exp(3)*2324 - q(6) = 15056;
a(7) = exp(3)*18332 - q(7) = 136652;
a(8) = exp(3)*167544 - q(8) = 1393722; ...
-
\p100 \\ set precision
{P=Vec(serlaplace(prod(k=1, 31, 1/(1-x^k/k +O(x^31))))); } \\ A007841
{Q=Vec(serlaplace(sum(n=1, 201, 3^n * prod(k=1, n, 1./(k-x^k +O(x^31)))))); }
for(n=0, 30, print1(round(exp(3)*P[n+1]-Q[n+1]), ", "))
A249477
E.g.f.: exp(4)*P(x) - Q(x), where P(x) = 1/Product_{n>=1} (1 - x^n/n) and Q(x) = Sum_{n>=1} 4^n/Product_{k=1..n} (k - x^k).
Original entry on oeis.org
1, 1, 7, 47, 360, 2884, 26068, 250140, 2659544, 30188024, 373401768, 4911407656, 69701336160, 1046114985408, 16770977757888, 283455401409920, 5076208319560320, 95434083840830080, 1890657361059194240, 39170792604756397440, 850920224456551054336, 19275340855527901297152
Offset: 0
E.g.f.: A(x) = 1 + x + 7*x^2/2! + 47*x^3/3! + 360*x^4/4! + 2884*x^5/5! +...
such that A(x) = exp(4)*P(x) - Q(x), where
P(x) = 1/Product_{n>=1} (1 - x^n/n) = Sum_{n>=0} A007841(n)*x^n/n!, and
Q(x) = Sum_{n>=1} 4^n / Product_{k=1..n} (k - x^k).
More explicitly,
P(x) = 1/((1-x)*(1-x^2/2)*(1-x^3/3)*(1-x^4/4)*(1-x^5/5)*...);
Q(x) = 4/(1-x) + 4^2/((1-x)*(2-x^2)) + 4^3/((1-x)*(2-x^2)*(3-x^3)) + 4^4/((1-x)*(2-x^2)*(3-x^3)*(4-x^4)) + 4^5/((1-x)*(2-x^2)*(3-x^3)*(4-x^4)*(5-x^5)) +...
We can illustrate the initial terms a(n) in the following manner.
The coefficients in Q(x) = Sum_{n>=0} q(n)*x^n/n! begin:
q(0) = 53.59815003314423907811...
q(1) = 53.59815003314423907811...
q(2) = 156.7944500994327172343...
q(3) = 553.5796503645866298592...
q(4) = 2697.496401856077388374...
q(5) = 14805.80061073873346130...
q(6) = 100818.1006770272116175...
q(7) = 750753.2864076001907799...
q(8) = 6488048.449153118392102...
q(9) = 61223693.06709220629587...
and the coefficients in P(x) = 1/Product_{n>=1} (1 - x^n/n) begin:
A007841 = [1, 1, 3, 11, 56, 324, 2324, 18332, 167544, ...];
from which we can generate this sequence like so:
a(0) = exp(4)*1 - q(0) = 1;
a(1) = exp(4)*1 - q(1) = 1;
a(2) = exp(4)*3 - q(2) = 7;
a(3) = exp(4)*11 - q(3) = 47;
a(4) = exp(4)*56 - q(4) = 360;
a(5) = exp(4)*324 - q(5) = 2884;
a(6) = exp(4)*2324 - q(6) = 26068;
a(7) = exp(4)*18332 - q(7) = 250140;
a(8) = exp(4)*167544 - q(8) = 2659544; ...
-
\p100 \\ set precision
{P=Vec(serlaplace(prod(k=1, 31, 1/(1-x^k/k +O(x^31))))); } \\ A007841
{Q=Vec(serlaplace(sum(n=1, 201, 4^n * prod(k=1, n, 1./(k-x^k +O(x^31)))))); }
for(n=0, 30, print1(round(exp(4)*P[n+1]-Q[n+1]), ", "))
A249478
E.g.f.: P(x)/exp(2) + Q(x), where P(x) = 1/Product_{n>=1} (1 - x^n/n) and Q(x) = Sum_{n>=1} -(-2)^n/Product_{k=1..n} (k - x^k).
Original entry on oeis.org
1, 1, 1, 5, 12, 88, 496, 4032, 32072, 335144, 3443928, 41477176, 523289472, 7298441952, 107525078304, 1714360202528, 28771306555776, 515446334184832, 9722819034952832, 193501572577378944, 4042243606465206784, 88584621284011603968, 2029364250844776170496, 48539531534286294782976
Offset: 0
E.g.f.: A(x) = 1 + x + x^2/2! + 5*x^3/3! + 12*x^4/4! + 88*x^5/5! +...
such that A(x) = exp(-2)*P(x) + Q(x), where
P(x) = 1/Product_{n>=1} (1 - x^n/n) = Sum_{n>=0} A007841(n)*x^n/n!, and
Q(x) = Sum_{n>=1} -(-2)^n / Product_{k=1..n} (k - x^k).
More explicitly,
P(x) = 1/((1-x)*(1-x^2/2)*(1-x^3/3)*(1-x^4/4)*(1-x^5/5)*...);
Q(x) = 2/(1-x) - 2^2/((1-x)*(2-x^2)) + 2^3/((1-x)*(2-x^2)*(3-x^3)) - 2^4/((1-x)*(2-x^2)*(3-x^3)*(4-x^4)) + 2^5/((1-x)*(2-x^2)*(3-x^3)*(4-x^4)*(5-x^5)) +...
We can illustrate the initial terms a(n) in the following manner.
The coefficients in Q(x) = Sum_{n>=0} q(n)*x^n/n! begin:
q(0) = 0.864664716763387308106000...
q(1) = 0.864664716763387308106000...
q(2) = 0.593994150290161924318001...
q(3) = 3.511311884397260389166005...
q(4) = 4.421224138749689253936028...
q(5) = 44.15136823133748782634416...
q(6) = 181.4808017581121040383451...
q(7) = 1551.033587706416132199201...
q(8) = 9397.385305404963149311748...
q(9) = 108557.0073471358880187848...
and the coefficients in P(x) = 1/Product_{n>=1} (1 - x^n/n) begin:
A007841 = [1, 1, 3, 11, 56, 324, 2324, 18332, 167544, ...];
from which we can generate this sequence like so:
a(0) = exp(-2)*1 + q(0) = 1;
a(1) = exp(-2)*1 + q(1) = 1;
a(2) = exp(-2)*3 + q(2) = 1;
a(3) = exp(-2)*11 + q(3) = 5;
a(4) = exp(-2)*56 + q(4) = 12;
a(5) = exp(-2)*324 + q(5) = 88;
a(6) = exp(-2)*2324 + q(6) = 496;
a(7) = exp(-2)*18332 + q(7) = 4032;
a(8) = exp(-2)*167544 + q(8) = 32072; ...
-
\p100 \\ set precision
{P=Vec(serlaplace(prod(k=1, 31, 1/(1-x^k/k +O(x^31))))); } \\ A007841
{Q=Vec(serlaplace(sum(n=1, 201, -(-2)^n * prod(k=1, n, 1./(k-x^k +O(x^31)))))); }
for(n=0, 30, print1(round(exp(-2)*P[n+1]+Q[n+1]), ", "))
A249590
E.g.f.: BesselI(0,2)*P(x) - Q(x), where P(x) = 1/Product_{n>=1} (1 - x^n/n^2) and Q(x) = Sum_{n>=1} 1/Product_{k=1..n} (k^2 - x^k).
Original entry on oeis.org
1, 1, 6, 63, 1162, 31263, 1207344, 61719326, 4103067834, 341454828363, 34946904263560, 4304483416099530, 629558493157805370, 107728435291299602135, 21346960361800584031800, 4847223770735591212039818, 1250978551922243595690043914, 364052135715732457875255719691
Offset: 0
E.g.f.: 1 + x + 6*x^2/2!^2 + 63*x^3/3!^2 + 1162*x^4/4!^2 + 31263*x^5/5!^2 +...
such that A(x) = BesselI(0,2)*P(x) - Q(x), where
P(x) = 1/Product_{n>=1} (1 - x^n/n^2) = Sum_{n>=0} A249588(n)*x^n/n!^2, and
Q(x) = Sum_{n>=1} 1/Product_{k=1..n} (k^2 - x^k).
More explicitly,
P(x) = 1/((1-x)*(1-x^2/4)*(1-x^3/9)*(1-x^4/16)*(1-x^5/25)*...);
Q(x) = 1/(1-x) + 1/((1-x)*(4-x^2)) + 1/((1-x)*(4-x^2)*(9-x^3)) + 1/((1-x)*(4-x^2)*(9-x^3)*(16-x^4)) + 1/((1-x)*(4-x^2)*(9-x^3)*(16-x^4)*(25-x^5)) +...
We can illustrate the initial terms a(n) in the following manner.
The coefficients q(n) in Q(x) = Sum_{n>=0} q(n)*x^n/n!^2 begin:
q(0) = 1.279585302336067267437204440811533...
q(1) = 1.279585302336067267437204440811533...
q(2) = 5.397926511680336337186022204057666...
q(3) = 48.69967981446729610442301759976513...
q(4) = 789.3250187996735809262470013346725...
q(5) = 19745.00072507184117617488656759887...
q(6) = 713288.6822890207712374724807435860...
q(7) = 34956701.28771539805703277298850790...
q(8) = 2239176303.370447012433955813571405...
q(9) = 181385849371.3820539848573249577420...
and the coefficients in P(x) = 1/Product_{n>=1} (1 - x^n/n^2) begin:
A249078 = [1, 1, 5, 49, 856, 22376, 842536, 42409480, 2782192064, ...];
from which we can generate this sequence like so:
a(0) = BesselI(0,2)*1 - q(0) = 1;
a(1) = BesselI(0,2)*1 - q(1) = 1;
a(2) = BesselI(0,2)*5 - q(2) = 6;
a(3) = BesselI(0,2)*49 - q(3) = 63;
a(4) = BesselI(0,2)*856 - q(4) = 1162;
a(5) = BesselI(0,2)*22376 - q(5) = 31263;
a(6) = BesselI(0,2)*842536 - q(6) = 1207344;
a(7) = BesselI(0,2)*42409480 - q(7) = 61719326;
a(8) = BesselI(0,2)*2782192064 - q(8) = 4103067834; ...
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\p100 \\ set precision
{P=Vec(serlaplace(serlaplace(prod(k=1, 31, 1/(1-x^k/k^2 +O(x^31)))))); } \\ A249588
{Q=Vec(serlaplace(serlaplace(sum(n=1, 201, prod(k=1, n, 1./(k^2-x^k +O(x^31))))))); }
for(n=0, 30, print1(round(besseli(0,2)*P[n+1]-Q[n+1]), ", "))
A249592
E.g.f.: exp(1)*P(x) - Q(x), where P(x) = 1/Product_{n>=1} (1 - x^n/n^2) and Q(x) = Sum_{n>=1} 1/Product_{k=1..n} (k - x^k/k).
Original entry on oeis.org
1, 1, 6, 64, 1192, 32360, 1257880, 64644520, 4315649600, 360332919360, 36979925855040, 4564758983929920, 668857835862650880, 114624254940995404800, 22742780483191398589440, 5169745984444274224143360, 1335478685859609449305006080, 388956774210908224056394014720
Offset: 0
E.g.f.: 1 + x + 6*x^2/2!^2 + 64*x^3/3!^2 + 1192*x^4/4!^2 + 32360*x^5/5!^2 +...
such that A(x) = exp(1)*P(x) - Q(x), where
P(x) = 1/Product_{n>=1} (1 - x^n/n^2) = Sum_{n>=0} A249588(n)*x^n/n!^2, and
Q(x) = Sum_{n>=1} 1/Product_{k=1..n} (k - x^k/k).
More explicitly,
P(x) = 1/((1-x)*(1-x^2/4)*(1-x^3/9)*(1-x^4/16)*(1-x^5/25)*...);
Q(x) = 1/(1-x) + 1/((1-x)*(2-x^2/2)) + 1/((1-x)*(2-x^2/2)*(3-x^3/3)) + 1/((1-x)*(2-x^2/2)*(3-x^3/3)*(4-x^4/4)) + 1/((1-x)*(2-x^2/2)*(3-x^3/3)*(4-x^4/4)*(5-x^5/5)) +...
We can illustrate the initial terms a(n) in the following manner.
The coefficients in Q(x) = Sum_{n>=0} q(n)*x^n/n! begin:
q(0) = 1.718281828459045235360287471352662...
q(1) = 1.718281828459045235360287471352662...
q(2) = 7.591409142295226176801437356763312...
q(3) = 69.19580959449321653265408609628046...
q(4) = 1134.849245160942721468406075477879...
q(5) = 28464.27419359959618642179245898717...
q(6) = 1032370.298622570136419515164963586...
q(7) = 50636398.83839730972810740431058131...
q(8) = 3247132530.854165002836403983556004...
q(9) = 263126229989.7260044371780752021631...
and the coefficients in P(x) = 1/Product_{n>=1} (1 - x^n/n^2) begin:
A007841 = [1, 1, 5, 49, 856, 22376, 842536, 42409480, 2782192064, ...];
from which we can generate this sequence like so:
a(0) = exp(1)*1 - q(0) = 1;
a(1) = exp(1)*1 - q(1) = 1;
a(2) = exp(1)*5 - q(2) = 6;
a(3) = exp(1)*49 - q(3) = 64;
a(4) = exp(1)*856 - q(4) = 1192;
a(5) = exp(1)*22376 - q(5) = 32360;
a(6) = exp(1)*842536 - q(6) = 1257880;
a(7) = exp(1)*42409480 - q(7) = 64644520;
a(8) = exp(1)*2782192064 - q(8) = 4315649600; ...
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\p100 \\ set precision
{P=Vec(serlaplace(serlaplace(prod(k=1, 31, 1/(1-x^k/k^2 +O(x^31)))))); } \\ A249588
{Q=Vec(serlaplace(serlaplace(sum(n=1, 201, prod(k=1, n, 1./(k-x^k/k +O(x^31))))))); }
for(n=0, 30, print1(round(exp(1)*P[n+1]-Q[n+1]), ", "))
Showing 1-9 of 9 results.
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