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A307419
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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.
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2
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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
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OFFSET
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0,5
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LINKS
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FORMULA
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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
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EXAMPLE
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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]
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MAPLE
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# 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
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MATHEMATICA
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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}];
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PROG
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(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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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