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 A165675 #41 Feb 27 2025 09:14:09
%S A165675 1,1,1,2,3,1,6,11,5,1,24,50,26,7,1,120,274,154,47,9,1,720,1764,1044,
%T A165675 342,74,11,1,5040,13068,8028,2754,638,107,13,1,40320,109584,69264,
%U A165675 24552,5944,1066,146,15,1,362880,1026576,663696,241128,60216,11274,1650,191,17,1
%N A165675 Triangle read by rows. T(n, k) = (n - k + 1)! * H(k, n - k), where H are the hyperharmonic numbers. For 0 <= k <= n.
%C A165675 Previous name: Extended triangle related to the asymptotic expansions of the E(x, m = 2, n).
%C A165675 For the definition of the hyperharmonic numbers see the formula section.
%C A165675 This triangle is the same as triangle A165674 except for the extra left-hand column T(n, 0) = n!. The T(n) formulas for the right-hand columns generate the coefficients of this extra left-hand column.
%C A165675 _Leroy Quet_ discovered triangle A105954 which is the reversal of our triangle.
%C A165675 In square format, row k gives the (n-1)-st elementary symmetric function of {k, k+1, k+2,..., k+n}, as in the Mathematica section. - _Clark Kimberling_, Dec 29 2011
%F A165675 The hyperharmonic numbers are H(n, k) = Sum_{j=0..k} H(n - 1, j), with base condition H(0, k) = 1/(k + 1).
%F A165675 T(n, k) = (n - k + 1)*T(n - 1, k) + T(n - 1, k - 1), 1 <= k <= n-1, with T(n, 0) = n! and T(n, n) = 1.
%F A165675 From _Peter Luschny_, Jul 03 2022: (Start)
%F A165675 The rectangular array is given by:
%F A165675 A(n, k) = (k + 1)!*H(n, k).
%F A165675 A(n, k) = (k + 1)*((n + k)! / n!)*hypergeom([-k, 1, 1], [2, n + 1], 1). (End)
%F A165675 From _Werner Schulte_, Feb 26 2025: (Start)
%F A165675 T(n, k) = n * T(n-1, k) + (n-1)! / (k-1)! for 0 < k < n.
%F A165675 T(n, k) = (Sum_{i=k..n} 1/i) * n! / (k-1)! for 0 < k <= n.
%F A165675 Matrix inverse M = T^(-1) is given by: M(n, n) = 1, M(n, n-1) = 1 - 2 * n for n > 0, M(n, n-2) = (n-1)^2 for n > 1, and M(i, j) = 0 otherwise. (End)
%e A165675 Triangle T(n, k) begins:
%e A165675 [0] 1;
%e A165675 [1] 1, 1;
%e A165675 [2] 2, 3, 1;
%e A165675 [3] 6, 11, 5, 1;
%e A165675 [4] 24, 50, 26, 7, 1;
%e A165675 [5] 120, 274, 154, 47, 9, 1;
%e A165675 [6] 720, 1764, 1044, 342, 74, 11, 1;
%e A165675 [7] 5040, 13068, 8028, 2754, 638, 107, 13, 1;
%e A165675 Seen as an array (the triangle arises when read by descending antidiagonals):
%e A165675 [0] 1, 1, 2, 6, 24, 120, 720, 5040, ...
%e A165675 [1] 1, 3, 11, 50, 274, 1764, 13068, 109584, ...
%e A165675 [2] 1, 5, 26, 154, 1044, 8028, 69264, 663696, ...
%e A165675 [3] 1, 7, 47, 342, 2754, 24552, 241128, 2592720, ...
%e A165675 [4] 1, 9, 74, 638, 5944, 60216, 662640, 7893840, ...
%e A165675 [5] 1, 11, 107, 1066, 11274, 127860, 1557660, 20355120, ...
%e A165675 [6] 1, 13, 146, 1650, 19524, 245004, 3272688, 46536624, ...
%e A165675 [7] 1, 15, 191, 2414, 31594, 434568, 6314664, 97053936, ...
%p A165675 nmax := 8; for n from 0 to nmax do a(n, 0) := n! od: for n from 0 to nmax do a(n, n) := 1 od: for n from 2 to nmax do for m from 1 to n-1 do a(n, m) := (n-m+1)*a(n-1, m) + a(n-1, m-1) od: od: seq(seq(a(n, m), m=0..n), n=0..nmax);
%p A165675 # _Johannes W. Meijer_, revised Nov 27 2012
%p A165675 # Shows the array format, using hyperharmonic numbers.
%p A165675 H := proc(n, k) option remember; if n = 0 then 1/(k+1)
%p A165675 else add(H(n - 1, j), j = 0..k) fi end:
%p A165675 seq(lprint(seq((k + 1)!*H(n, k), k = 0..7)), n = 0..7);
%p A165675 # Shows the array format, using the hypergeometric formula.
%p A165675 A := (n, k) -> (k+1)*((n + k)! / n!)*hypergeom([-k, 1, 1], [2, n + 1], 1):
%p A165675 seq(lprint(seq(simplify(A(n, k)), k = 0..7)), n = 0..7);
%p A165675 # _Peter Luschny_, Jul 03 2022
%t A165675 a[n_] := SymmetricPolynomial[n - 1, t[n]]; z = 10;
%t A165675 t[n_] := Table[k - 1, {k, 1, n}]; t1 = Table[a[n], {n, 1, z}] (* A000142 *)
%t A165675 t[n_] := Table[k, {k, 1, n}]; t2 = Table[a[n], {n, 1, z}] (* A000254 *)
%t A165675 t[n_] := Table[k + 1, {k, 1, n}]; t3 = Table[a[n], {n, 1, z}] (* A001705 *)
%t A165675 t[n_] := Table[k + 2, {k, 1, n}]; t4 = Table[a[n], {n, 1, z}] (* A001711 *)
%t A165675 t[n_] := Table[k + 3, {k, 1, n}]; t5 = Table[a[n], {n, 1, z}] (* A001716 *)
%t A165675 t[n_] := Table[k + 4, {k, 1, n}]; t6 = Table[a[n], {n, 1, z}] (* A001721 *)
%t A165675 t[n_] := Table[k + 5, {k, 1, n}]; t7 = Table[a[n], {n, 1, z}] (* A051524 *)
%t A165675 t[n_] := Table[k + 6, {k, 1, n}]; t8 = Table[a[n], {n, 1, z}] (* A051545 *)
%t A165675 t[n_] := Table[k + 7, {k, 1, n}]; t9 = Table[a[n], {n, 1, z}] (* A051560 *)
%t A165675 t[n_] := Table[k + 8, {k, 1, n}]; t10 = Table[a[n], {n, 1, z}] (* A051562 *)
%t A165675 t[n_] := Table[k + 9, {k, 1, n}]; t11 = Table[a[n], {n, 1, z}] (* A051564 *)
%t A165675 t[n_] := Table[k + 10, {k, 1, n}];t12 = Table[a[n], {n, 1, z}] (* A203147 *)
%t A165675 t = {t1, t2, t3, t4, t5, t6, t7, t8, t9, t10};
%t A165675 TableForm[t] (* A165675 in square format *)
%t A165675 m[i_, j_] := t[[i]][[j]];
%t A165675 (* A165675 as a sequence *)
%t A165675 Flatten[Table[m[i, n + 1 - i], {n, 1, 10}, {i, 1, n}]]
%t A165675 (* _Clark Kimberling_, Dec 29 2011 *)
%t A165675 A[n_, k_] := (k + 1)*((n + k)! / n!)*HypergeometricPFQ[{-k, 1, 1}, {2, n + 1}, 1];
%t A165675 Table[A[n, k], {n, 0, 7}, {k, 0, 7}] // TableForm (* _Peter Luschny_, Jul 03 2022 *)
%o A165675 (Python)
%o A165675 from functools import cache
%o A165675 @cache
%o A165675 def Trow(n: int) -> list[int]:
%o A165675 if n == 0:
%o A165675 return [1]
%o A165675 row = Trow(n - 1) + [1]
%o A165675 for m in range(n - 1, 0, -1):
%o A165675 row[m] = (n - m + 1) * row[m] + row[m - 1]
%o A165675 row[0] *= n
%o A165675 return row
%o A165675 for n in range(9): print(Trow(n)) # _Peter Luschny_, Feb 27 2025
%Y A165675 A105954 is the reversal of this triangle.
%Y A165675 A165674, A138771 and A165680 are related triangles.
%Y A165675 A080663 equals the third right hand column.
%Y A165675 A000142 equals the first left hand column.
%Y A165675 A093345 are the row sums.
%Y A165675 Columns include A165676, A165677, A165678 and A165679.
%K A165675 easy,nonn,tabl
%O A165675 0,4
%A A165675 _Johannes W. Meijer_, Oct 05 2009
%E A165675 New name from _Peter Luschny_, Jul 03 2022