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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.

%I A307419 #46 Dec 05 2024 19:16:15
%S A307419 1,0,1,0,3,1,0,11,9,1,0,50,71,18,1,0,274,580,245,30,1,0,1764,5104,
%T A307419 3135,625,45,1,0,13068,48860,40369,11515,1330,63,1,0,109584,509004,
%U A307419 537628,203889,33320,2506,84,1,0,1026576,5753736,7494416,3602088,775929,81900,4326,108,1
%N 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.
%F A307419 E.g.f.: A(t,x) = (1-t)^(-x/(1-t)).
%F A307419 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.
%F A307419 T(n, k) = n!*Sum_{i=0..n-k} abs(Stirling1(n-i, k))/(n-i)!*binomial(i+k-1, i).
%F A307419 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
%F A307419 T(n, k) = Sum_{j=k..n} (-1)^(n-j)*binomial(j, k)*Stirling1(n, j)*k^(j-k). - _Peter Luschny_, Jun 09 2022
%e A307419 Triangle starts:
%e A307419 0: [1]
%e A307419 1: [0,       1]
%e A307419 2: [0,       3,       1]
%e A307419 3: [0,      11,       9,       1]
%e A307419 4: [0,      50,      71,      18,       1]
%e A307419 5: [0,     274,     580,     245,      30,      1]
%e A307419 6: [0,    1764,    5104,    3135,     625,     45,     1]
%e A307419 7: [0,   13068,   48860,   40369,   11515,   1330,    63,    1]
%e A307419 8: [0,  109584,  509004,  537628,  203889,  33320,  2506,   84,   1]
%e A307419 9: [0, 1026576, 5753736, 7494416, 3602088, 775929, 81900, 4326, 108, 1]
%e A307419 Col:   A000254, A001706, A001713, A001719, ...
%p A307419 # Note that for n > 16 Maple fails (at least in some versions) to compute the
%p A307419 # terms properly. Inserting 'simplify' or numerical evaluation might help.
%p A307419 A307419Row := proc(n) local ogf, ser; ogf := (n, k) -> GAMMA(n+k+x)/GAMMA(k+x);
%p A307419 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);
%p A307419 # Alternatively by the egf for column k:
%p A307419 A307419Col := proc(n, len) local f, egf, ser; f := (n,x) -> (log(1-x)/(x-1))^n/n!;
%p A307419 egf := (n,x) -> diff(f(n, x), [x$n]); ser := n -> series(egf(n, x), x, len);
%p A307419 seq(k!*coeff(ser(n), x, k), k=0..len-1) end:
%p A307419 seq(print(A307419Col(k, 10)), k=0..9); # _Peter Luschny_, Apr 12 2019
%p A307419 T := (n, k) -> add((-1)^(n-j)*binomial(j, k)*Stirling1(n, j)*k^(j-k), j = k..n):
%p A307419 seq(seq(T(n,k), k = 0..n), n = 0..9); # _Peter Luschny_, Jun 09 2022
%t A307419 f[n_, x_] := f[n, x] = D[(Log[1 - x]/(x - 1))^n/n!, {x, n}];
%t A307419 T[n_, k_] := (n - k)! SeriesCoefficient[f[k, x], {x, 0, n - k}];
%t A307419 Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 13 2019 *)
%o A307419 (Maxima) T(n,k):=n!*sum((binomial(k+i-1,i)*abs(stirling1(n-i,k)))/(n-i)!,i,0,n-k);
%o A307419 (Maxima) taylor((1-t)^(-x/(1-t)),t,0,7,x,0,7);
%o A307419 (Maxima) T(n,k):=coeff(taylor(gamma(n+k+t)/gamma(k+t),t,0,10),t,k);
%o A307419 (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
%Y A307419 Row sums are A087761.
%Y A307419 Columns are A000007, A000254, A001706, A001713, A001719.
%Y A307419 Cf. A001008/A002805.
%K A307419 nonn,tabl
%O A307419 0,5
%A A307419 _Vladimir Kruchinin_, Apr 08 2019