A320253
Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 + k - Sum_{j=1..k} exp(j*x)).
Original entry on oeis.org
1, 1, 0, 1, 1, 0, 1, 3, 3, 0, 1, 6, 23, 13, 0, 1, 10, 86, 261, 75, 0, 1, 15, 230, 1836, 3947, 541, 0, 1, 21, 505, 7900, 52250, 74613, 4683, 0, 1, 28, 973, 25425, 361754, 1858716, 1692563, 47293, 0, 1, 36, 1708, 67473, 1706629, 20706700, 79345346, 44794221, 545835, 0
Offset: 0
E.g.f. of column k: A_k(x) = 1 + (1/2)*k*(k + 1)*x/1! + (1/6)*k*(3*k^3 + 8*k^2 + 6*k + 1)*x^2/2! + (1/4)*k^2*(k + 1)^2*(3*k^2 + 7*k + 3)*x^3/3! + (1/30)*k*(45*k^7 + 270*k^6 + 635*k^5 + 741*k^4 + 440*k^3 + 115*k^2 + 5*k - 1)*x^4/4! + ...
Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 1, 3, 6, 10, 15, ...
0, 3, 23, 86, 230, 505, ...
0, 13, 261, 1836, 7900, 25425, ...
0, 75, 3947, 52250, 361754, 1706629, ...
0, 541, 74613, 1858716, 20706700, 143195025, ...
Columns k=0..10 give
A000007,
A000670,
A004700,
A004701,
A004702,
A004703,
A004704,
A004705,
A004706,
A004707,
A004708.
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Table[Function[k, n! SeriesCoefficient[1/(1 + k - Sum[Exp[i x], {i, 1, k}]), {x, 0, n}]][j - n], {j, 0, 9}, {n, 0, j}] // Flatten
Table[Function[k, n! SeriesCoefficient[1/(1 + k - Exp[x] (Exp[k x] - 1)/(Exp[x] - 1)), {x, 0, n}]][j - n], {j, 0, 9}, {n, 0, j}] // Flatten
A319508
a(n) = n! * [x^n] 1/(1 + n - exp(x)*(exp(n*x) - 1)/(exp(x) - 1)).
Original entry on oeis.org
1, 1, 23, 1836, 361754, 143195025, 99986786773, 112625837135056, 191736660977760804, 469456525723134676365, 1589874326596159958849175, 7216642860485686755145923828, 42781019992428263086709058587150, 324097110833947198922869762652717041
Offset: 0
Cf.
A000670,
A004700,
A004701,
A004702,
A004703,
A004704,
A004705,
A004706,
A004707,
A004708,
A319509.
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Table[n! SeriesCoefficient[1/(1 + n - Exp[x] (Exp[n x] - 1)/(Exp[x] - 1)), {x, 0, n}], {n, 0, 13}]
-
default(seriesprecision, 101); {a(n) = n!*polcoeff((1/(1+n-exp(x)*(exp(n*x)-1)/(exp(x)-1)) + O(x^(n+1))), n)};
for(n=0, 15, print1(a(n), ", ")) \\ G. C. Greubel, Oct 09 2018
A355410
Expansion of e.g.f. 1/(3 - exp(x) - exp(3*x)).
Original entry on oeis.org
1, 4, 42, 652, 13482, 348484, 10809282, 391162972, 16177467642, 752689508404, 38911563009522, 2212759299753292, 137270821971529002, 9225382887659221924, 667690580181890112162, 51776098497454677943612, 4282645413209764715753562
Offset: 0
-
A355410 := proc(n)
option remember ;
if n = 0 then
1;
else
add((3^k + 1) * binomial(n,k) * procname(n-k),k=1..n) ;
end if;
end proc:
seq(A355410(n),n=0..70) ; # R. J. Mathar, Dec 04 2023
-
my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(3-exp(x)-exp(3*x))))
-
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, (3^j+1)*binomial(i, j)*v[i-j+1])); v;
A355425
Expansion of e.g.f. 1/(1 - Sum_{k=1..2} (exp(k*x) - 1)/k).
Original entry on oeis.org
1, 2, 11, 89, 959, 12917, 208781, 3937019, 84846899, 2057107337, 55416031601, 1642126375199, 53084324076839, 1859037341680157, 70112365228588421, 2833115932639555379, 122113252334984094779, 5592296493425013663377, 271169701559687033317241
Offset: 0
-
my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=1, 2, (exp(k*x)-1)/k))))
-
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, (1+2^(j-1))*binomial(i, j)*v[i-j+1])); v;
A368013
Expansion of e.g.f. 1/(1 - exp(x) + exp(2*x)).
Original entry on oeis.org
1, -1, -1, 5, 11, -91, -301, 3485, 15371, -228811, -1261501, 22951565, 151846331, -3264973531, -25201039501, 625232757245, 5515342166891, -155079142742251, -1538993024478301, 48364005482108525, 533289474412481051, -18523127502677822971
Offset: 0
-
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, (1-2^j)*binomial(i, j)*v[i-j+1])); v;
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my(x='x+O('x^30)); Vec(serlaplace(1/(1 - exp(x) + exp(2*x)))) \\ Joerg Arndt, Feb 10 2025
Showing 1-5 of 5 results.