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.
-
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.
-
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
A366298
Expansion of e.g.f. 1 / (-2 + Sum_{k=1..3} exp(-k*x)).
Original entry on oeis.org
1, 6, 58, 828, 15766, 375276, 10719118, 357202068, 13603819126, 582854637276, 27747071520478, 1453003753611108, 83005119616449286, 5136947527401250476, 342365553703113120238, 24447711909762202272948, 1862151878019906517540246, 150702660087903415402794876, 12913688931657425188926182398
Offset: 0
-
nmax = 18; CoefficientList[Series[1/(-2 + Sum[Exp[-k x], {k, 1, 3}]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[(-1)^(k + 1) Binomial[n, k] (1 + 2^k + 3^k) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
A355421
Expansion of e.g.f. exp(Sum_{k=1..3} (exp(k*x) - 1)).
Original entry on oeis.org
1, 6, 50, 504, 5870, 76872, 1111646, 17522664, 298133054, 5433157512, 105396184478, 2165189912040, 46901678992958, 1067332196912136, 25435754924426270, 633014456504059368, 16411191933603611198, 442258823578968351624
Offset: 0
-
my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, 3, exp(k*x)-1))))
-
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+3^j)*binomial(i-1, j-1)*v[i-j+1])); v;
A355426
Expansion of e.g.f. 1/(1 - Sum_{k=1..3} (exp(k*x) - 1)/k).
Original entry on oeis.org
1, 3, 24, 284, 4476, 88178, 2084564, 57493334, 1812223276, 64262620538, 2531993864004, 109738634393534, 5188538157065276, 265761817180172498, 14659691726110341844, 866403731832477234134, 54619096812884242006476, 3658454458052874579886058
Offset: 0
-
my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=1, 3, (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)+3^(j-1))*binomial(i, j)*v[i-j+1])); v;
Showing 1-5 of 5 results.