A249078
E.g.f.: exp(1)*P(x) - Q(x), where P(x) = 1/Product_{n>=1} (1 - x^n/n) and Q(x) = Sum_{n>=1} 1/Product_{k=1..n} (k - x^k).
Original entry on oeis.org
1, 1, 4, 17, 96, 595, 4516, 37104, 351020, 3604001, 41007240, 502039444, 6703536516, 95376507135, 1459072099824, 23677731306350, 408821193129564, 7443839953433701, 143258713990271960, 2893053522512463984, 61396438056305204020, 1362146168353191078195, 31605702195327725326560
Offset: 0
E.g.f.: A(x) = 1 + x + 4*x^2/2! + 17*x^3/3! + 96*x^4/4! + 595*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/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) = 1.7182818284590452...
q(1) = 1.7182818284590452...
q(2) = 4.1548454853771357...
q(3) = 12.901100113049497...
q(4) = 56.223782393706533...
q(5) = 285.72331242073065...
q(6) = 1801.2869693388211...
q(7) = 12727.542479311217...
q(8) = 104411.81066734227...
q(9) = 947120.40724315491...
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) = 4;
a(3) = exp(1)*11 - q(3) = 17;
a(4) = exp(1)*56 - q(4) = 96;
a(5) = exp(1)*324 - q(5) = 595;
a(6) = exp(1)*2324 - q(6) = 4516;
a(7) = exp(1)*18332 - q(7) = 37104;
a(8) = exp(1)*167544 - q(8) = 351020; ...
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\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,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]),", "))
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; ...
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\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; ...
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\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; ...
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\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]), ", "))
Showing 1-5 of 5 results.
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