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.
-
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
A366299
Expansion of e.g.f. 1 / (-3 + Sum_{k=1..4} exp(-k*x)).
Original entry on oeis.org
1, 10, 170, 4300, 145046, 6115900, 309453710, 18267444100, 1232400398966, 93535914320620, 7887919177776350, 731710341934820500, 74046493229735962886, 8117679564133907097340, 958393800813241073719790, 121232569802975799394430500, 16357741845227058108680934806, 2345072789674603792983906178060
Offset: 0
-
nmax = 17; CoefficientList[Series[1/(-3 + Sum[Exp[-k x], {k, 1, 4}]), {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 + 4^k) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 17}]
A355422
Expansion of e.g.f. exp(Sum_{k=1..4} (exp(k*x) - 1)).
Original entry on oeis.org
1, 10, 130, 2000, 35054, 684000, 14628190, 338990000, 8438270014, 224070580800, 6311530677150, 187702155610000, 5870416574854974, 192423935736656800, 6591135679171866910, 235315671951948070000, 8736534653549465359934
Offset: 0
-
my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, 4, 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+4^j)*binomial(i-1, j-1)*v[i-j+1])); v;
Showing 1-4 of 4 results.