A307419 Triangle of harmonic numbers T(n, k) = [t^n] Gamma(n+k+t)/Gamma(k+t) for n >= 0 and 0 <= k <= n, read by rows.

1, 0, 1, 0, 3, 1, 0, 11, 9, 1, 0, 50, 71, 18, 1, 0, 274, 580, 245, 30, 1, 0, 1764, 5104, 3135, 625, 45, 1, 0, 13068, 48860, 40369, 11515, 1330, 63, 1, 0, 109584, 509004, 537628, 203889, 33320, 2506, 84, 1, 0, 1026576, 5753736, 7494416, 3602088, 775929, 81900, 4326, 108, 1

Offset

0,5

Links

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

Formula

E.g.f.: ((1-t)^(-x/(1-t)).

T(n, k) = n!*Sum_{L1+L2+...+Lk=n} H(L1)H(L2)...H(Lk) with Li > 0, where H(n) are the harmonic numbers A001008.

T(n, k) = n!*Sum_{i=0..n-k} abs(Stirling1(n-i, k))/(n-i)!*binomial(i+k-1, i).

T(n, k) = k! [x^k] (d^n/dx^n) ((log(1-x)/(x-1))^n/n!), the e.g.f. for column k where Col(k) = [T(n+k, k) for n = 0, 1, 2, ...]. - Peter Luschny, Apr 12 2019

T(n, k) = Sum_{j=k..n} (-1)^(n-j)*binomial(j, k)*Stirling1(n, j)*k^(j-k). - Peter Luschny, Jun 09 2022

Example

Triangle starts:
0: [1]
1: [0,       1]
2: [0,       3,       1]
3: [0,      11,       9,       1]
4: [0,      50,      71,      18,       1]
5: [0,     274,     580,     245,      30,      1]
6: [0,    1764,    5104,    3135,     625,     45,     1]
7: [0,   13068,   48860,   40369,   11515,   1330,    63,    1]
8: [0,  109584,  509004,  537628,  203889,  33320,  2506,   84,   1]
9: [0, 1026576, 5753736, 7494416, 3602088, 775929, 81900, 4326, 108, 1]
Col:   A000254, A001706, A001713, A001719, ...

Maple

# Note that for n > 16 Maple fails (at least in some versions) to compute the
# terms properly. Inserting 'simplify' or numerical evaluation might help.
A307419Row := proc(n) local ogf, ser; ogf := (n, k) -> GAMMA(n+k+x)/GAMMA(k+x);
ser := (n, k) -> series(ogf(n, k), x, k+2); seq(coeff(ser(n, k), x, k), k=0..n) end: seq(A307419Row(n), n=0..9);
# Alternatively by the egf for column k:
A307419Col := proc(n, len) local f, egf, ser; f := (n, x) -> (log(1-x)/(x-1))^n/n!;
egf := (n, x) -> diff(f(n, x), [x$n]); ser := n -> series(egf(n, x), x, len);
seq(k!*coeff(ser(n), x, k), k=0..len-1) end:
seq(print(A307419Col(k, 10)), k=0..9); # Peter Luschny, Apr 12 2019
T := (n, k) -> add((-1)^(n-j)*binomial(j, k)*Stirling1(n, j)*k^(j-k), j = k..n):
seq(seq(T(n, k), k = 0..n), n = 0..9); # Peter Luschny, Jun 09 2022

Mathematica

f[n_, x_] := f[n, x] = D[(Log[1 - x]/(x - 1))^n/n!, {x, n}];
T[n_, k_] := (n - k)! SeriesCoefficient[f[k, x], {x, 0, n - k}];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 13 2019 *)

Prog

(Maxima) T(n, k):=n!*sum((binomial(k+i-1, i)*abs(stirling1(n-i, k)))/(n-i)!, i, 0, n-k)
(Maxima) taylor((1-t)^(-x/(1-t)), t, 0, 7, x, 0, 7);
(Maxima) T(n, k):=coeff(taylor(gamma(n+k+t)/gamma(k+t), t, 0, 10), t, k);
(PARI) T(n, k) = n!*sum(i=0, n-k, abs(stirling(n-i, k, 1))*binomial(i+k-1, i)/(n-i)!); \\ Michel Marcus, Apr 13 2019

Crossrefs

Row sums are A087761.

Columns are A000007, A000254, A001706, A001713, A001719.

Cf. A001008/A002805.

Sequence in context: A067176 A249480 A271704 * A256892 A256893 A359759

Adjacent sequences: A307416 A307417 A307418 * A307420 A307421 A307422

Keyword

nonn,tabl

Author

Vladimir Kruchinin, Apr 08 2019

Status

approved