A004708 -id:A004708

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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)).

+10
10

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,8

LINKS

Table of n, a(n) for n=0..54.

FORMULA

E.g.f. of column k: 1/(1 + k - exp(x)*(exp(k*x) - 1)/(exp(x) - 1)).

EXAMPLE

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, ...

MATHEMATICA

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

CROSSREFS

Columns k=0..10 give A000007, A000670, A004700, A004701, A004702, A004703, A004704, A004705, A004706, A004707, A004708.

Main diagonal gives A319508.

KEYWORD

nonn,tabl

AUTHOR

Ilya Gutkovskiy, Oct 08 2018

STATUS

approved

A319508

a(n) = n! * [x^n] 1/(1 + n - exp(x)*(exp(n*x) - 1)/(exp(x) - 1)).

+10
8

1, 1, 23, 1836, 361754, 143195025, 99986786773, 112625837135056, 191736660977760804, 469456525723134676365, 1589874326596159958849175, 7216642860485686755145923828, 42781019992428263086709058587150, 324097110833947198922869762652717041

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..167

FORMULA

a(n) = n! * [x^n] 1/(1 + n - exp(x) - exp(2*x) - exp(3*x) - ... - exp(n*x)).

a(n) ~ sqrt(2*Pi) * n^(3*n + 1/2) / (2^n * exp(n - 5/3)). - Vaclav Kotesovec, Oct 09 2018

MATHEMATICA

Table[n! SeriesCoefficient[1/(1 + n - Exp[x] (Exp[n x] - 1)/(Exp[x] - 1)), {x, 0, n}], {n, 0, 13}]

PROG

(PARI) 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

CROSSREFS

Main diagonal of A320253.

Cf. A000670, A004700, A004701, A004702, A004703, A004704, A004705, A004706, A004707, A004708, A319509.

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Sep 21 2018

STATUS

approved

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