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A039816 Triangle read by rows: matrix 4th power of the Stirling-1 triangle A008275.

%I A039816 #32 Sep 18 2022 12:36:37
%S A039816 1,-4,1,26,-12,1,-234,152,-24,1,2696,-2210,500,-40,1,-37919,36976,
%T A039816 -10710,1240,-60,1,630521,-704837,245896,-36750,2590,-84,1,-12111114,
%U A039816 15132932,-6120324,1109696,-101500,4816,-112,1,264051201,-362099010,165387680,-34990620,3901296,-241164,8232,-144,1
%N A039816 Triangle read by rows: matrix 4th power of the Stirling-1 triangle A008275.
%H A039816 Seiichi Manyama, <a href="/A039816/b039816.txt">Rows n = 1..140, flattened</a>
%H A039816 Gabriella Bretti, Pierpaolo Natalini and Paolo E. Ricci, <a href="https://doi.org/10.1515/gmj-2019-2007">A new set of Sheffer-Bell polynomials and logarithmic numbers</a>, Georgian Mathematical Journal, Feb. 2019, page 9.
%F A039816 E.g.f. of k-th column: ((log(1+log(1+log(1+log(1+x)))))^k)/k!.
%e A039816 Triangle begins:
%e A039816        1;
%e A039816       -4,     1;
%e A039816       26,   -12,      1;
%e A039816     -234,   152,    -24,    1;
%e A039816     2696, -2210,    500,  -40,   1;
%e A039816   -37919, 36976, -10710, 1240, -60, 1;
%e A039816   ...
%p A039816 T:= Matrix(10,10,(i,j) -> `if`(i>= j, combinat:-stirling1(i,j),0)):
%p A039816 M:= T^4:
%p A039816 seq(seq(M[i,j],j=1..i),i=1..10); # _Robert Israel_, Sep 12 2022
%t A039816 Flatten[Table[SeriesCoefficient[(Log[1+Log[1+Log[1+Log[1+x]]]])^k,{x,0,n}] n!/k!, {n,9}, {k,n}]] (* _Stefano Spezia_, Sep 12 2022 *)
%Y A039816 Cf. A000310 (first column), A008275.
%Y A039816 Cf. A039812, A039814, A039815, A039817.
%K A039816 sign,tabl
%O A039816 1,2
%A A039816 _Christian G. Bower_, Feb 15 1999