This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A192001 #41 Sep 23 2022 23:59:23
%S A192001 1,2,1,3,3,1,4,6,5,1,5,10,14,9,1,6,15,30,36,17,1,7,21,55,100,98,33,1,
%T A192001 8,28,91,225,354,276,65,1,9,36,140,441,979,1300,794,129,1,10,45,204,
%U A192001 784,2275,4425,4890,2316,257,1
%N A192001 Triangle with sums of nonnegative integer powers of positive first n integers in the columns.
%C A192001 This is the Abramowitz-Stegun table on p. 813, call it s(m,n), with an extra column n=0 with values n added, and read by antidiagonals. a(n,m) = s(n-m,m), n+1 >= m >= 0.
%C A192001 O.g.f. for column no. m >= 0: (x^(m+1)/(1-x)^(m+2))*E(m;x) with the row polynomials E(m;x) = Sum_{p=0..m} A173018(m,p)*x^p of the Eulerian number triangle (proof via the Worpitzky identity). See the Graham et al. reference p. 253-8 for Eulerian numbers, and the Worpitzky identity (6.37) on p. 255.
%C A192001 E.g.f. for diagonals (starting with k=0 for the main diagonal): g(k,x) = exp(x)*(exp((k+1)*x)-1)/(1-exp(x)).
%C A192001 Compare with (7.77) on p. 353 of the Graham et al. reference.
%C A192001 O.g.f. for diagonals (starting with k=0 for the main diagonal): G(k,z) =(Psi(1/z+1)-Psi(1/z-k-1))/z - 1.
%C A192001 with the digamma function Psi(z):=(log(Gamma(z)))'.
%C A192001 Compare with Graham et al., p. 352, eq.(7.76), where H_z=Psi(z+1)+gamma, with the Euler-Mascheroni constant gamma.
%C A192001 The diagonal sequences are, for k=0..9: A000012, A000051, A001550-A001557.
%C A192001 The negative k-diagonal, -a(k+m+1,m), yields the Sheffer z-sequence Shz(k+1;m) for the Sheffer arrays |S1|(k+1) defined in a comment to A094646.
%C A192001 See also A196837 with a W. Lang link, where the o.g.f.s for the diagonals, numbered with k >= 1, are given as G(k,x) = Sum_{m=0..k} (k-m)*S1(k+1,k+1-m)*x^m / Product_{j=1..k} (1-j*x), with S1 the Stirling numbers of the first kind, A048994. - _Wolfdieter Lang_, Nov 01 2011
%D A192001 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964. Tenth printing, Wiley, 2002 (also electronically available, see the link), p. 813.
%D A192001 Mohammad K. Azarian, Problem 1218, Pi Mu Epsilon Journal, Vol. 13, No. 2, Spring 2010, p. 116. Solution published in Vol. 13, No. 3, Fall 2010, pp. 183-185.
%D A192001 Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, Addison-Wesley, 1991.
%H A192001 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [scanned copy], p. 813.
%F A192001 a(n,m) = s(n-m,m), n-1 >= m >= 0, n >= 1, else 0, with s(n,m) := Sum_{k=1..n} k^m.
%F A192001 O.g.f. column no. m: see a comment above.
%F A192001 O.g.f.s and e.g.f.s for diagonals k >= 0: see a comment above.
%F A192001 Recurrence known by Ibn al-Haytham (see a comment and link under A000537, and comments by _Bruno Berselli_ under the A-numbers of the first column sequences):
%F A192001 a(n,m) = (n-m)*a(n-1,m-1) - Sum_{j=m..n-2} a(j,m-1), n >= 1, n-1 >= m >= 1. a(n,0) = n. - _Wolfdieter Lang_, Jan 12 2013
%e A192001 The triangle a(n,m) begins:
%e A192001 n\m 0 1 2 3 4 5 6 7 8 9 10 ...
%e A192001 n=1: 1
%e A192001 n=2: 2 1
%e A192001 n=3: 3 3 1
%e A192001 n=4: 4 6 5 1
%e A192001 n=5: 5 10 14 9 1
%e A192001 n=6: 6 15 30 36 17 1
%e A192001 n=7: 7 21 55 100 98 33 1
%e A192001 n=8: 8 28 91 225 354 276 65 1
%e A192001 n=9: 9 36 140 441 979 1300 794 129 1
%e A192001 n=10: 10 45 204 784 2275 4425 4890 2316 257 1
%e A192001 n=11: 11 55 285 1296 4676 12201 20515 18700 6818 513 1
%e A192001 ... Reformatted and extended by _Wolfdieter Lang_, Jan 12 2013
%e A192001 a(4,2)= 5 = s(2,2) = 1^2 + 2^2.
%e A192001 Recurrence: 55 = a(7,2) = (7-2)*a(6,1) - (a(2,1) + a(3,1) + a(4,1) + a(5,1)) = 5*15 - (1 + 3 + 6 + 10) = 55. - _Wolfdieter Lang_, Jan 12 2013
%e A192001 The first column, m=0 holds the integers 1,2,3,..., equal to the sums of 0th powers of the n first integers. The second column is 1, 1+2, 1+2+3, ... = A000217. The third column are the sums of squares, 1^2, 1^2+2^2, 1^2+2^2+3^3, ... = A000330, etc. - _M. F. Hasler_, Jan 13 2013
%t A192001 Flatten[ Table[ HarmonicNumber[-m + n, -m], {n, 1, 10}, {m, 0, n - 1}]] (* _Jean-François Alcover_, Sep 26 2011 *)
%o A192001 (PARI) A192001(n,m) = sum(k=1,n-m,k^m) \\ - _M. F. Hasler_, Jan 13 2013
%Y A192001 Cf. A103438 (omitting the first column of zeros, reversed).
%K A192001 nonn,easy,tabl
%O A192001 1,2
%A A192001 _Wolfdieter Lang_, Jun 25 2011