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A125166 Triangle R(n,k), companion to A125165, left column n^3.

%I A125166 #41 May 05 2025 23:41:01
%S A125166 1,8,1,27,9,1,64,36,10,1,125,100,46,11,1,216,225,146,57,12,1,343,441,
%T A125166 371,203,69,13,1,512,784,812,574,272,82,14,1,729,1296,1596,1386,846,
%U A125166 354,96,15,1,1000,2025,2892,2982,2232,1200,450,111,16,1
%N A125166 Triangle R(n,k), companion to A125165, left column n^3.
%C A125166 Riordan array ((1 + 4*x + x^2)/(1 - x)^4, x/(1 - x)). - _Philippe Deléham_, Dec 09 2013
%H A125166 Paolo Xausa, <a href="/A125166/b125166.txt">Table of n, a(n) for n = 0..11475</a> (rows 0..150 of triangle, flattened).
%F A125166 Binomial transform of an infinite matrix M with diagonal D, subdiagonal (D-1)..., etc; as follows: (D) = (1,1,1...); (D-1) = (7,7,7...); (D-2) = (12,12,12...); (D-3) = (6,6,6...). Alternatively, given left border n^3: (1, 8, 27, 64...); for k>1, R(n,k) = R(n-1,k) + R(n-1,k-1).
%F A125166 From _Wolfdieter Lang_, Mar 27 2025: (Start)
%F A125166 Riordan triangle (see a comment above):
%F A125166 R(n, 0) = (n+1)^3, R(n, k) = R(n-1, k-1) + R(n-1, k), for k >= 1. (from the (finite) A-sequence {1, 1} with offset 0),
%F A125166 R(n, 0) = Sum_{k=0..n-1} Z(j)*R(n-1, k), for n >= 1, and R(0, 0) = 1, with the Riordan Z-sequence A382057. For Riordan A- and Z-sequences see the first W. Lang link in A006232.
%F A125166 R(n, k) = Sum{j=0..n} (j+1)^3*A097805(n-j, k) (convolution property).
%F A125166 R(n, k) = Sum{j=0..3} A028246(4, j+1)*binomial(n, k+j). (Proof for k=0 with a standard (n+1)^3  formula, and for k >= 1 using the Pascal type recurrence for the triangle.)
%F A125166 O.g.f. column k (with leading 0s): ((1 + 4*x + x^2)/(1 - x)^4)*(x/(1-x))^k. (Numerator polynomial from row 3 of the triangle A008292.)
%F A125166 O.g.f. row polynomials P(n, y) = Sum_{k=0..n} R(n, k)*y^k:
%F A125166    G(y, x) = (1 + 4*x + x^2)/((1 - x)^3*(1 - (1+y)*x)). (End)
%e A125166 With other offset (k >= 1) from first formula: R(5,3) = 146 = R(4,3) + R(4,2) = 46 + 100.
%e A125166 The Riordan triangle R begins:
%e A125166   n\k|     0    1    2    3    4    5   6   7   8  9
%e A125166   --------------------------------------------------
%e A125166    0 |     1
%e A125166    1 |     8    1
%e A125166    2 |    27    9    1
%e A125166    3 |    64   36   10    1
%e A125166    4 |   125  100   46   11    1
%e A125166    5 |   216  225  146   57   12    1
%e A125166    6 |   343  441  371  203   69   13   1
%e A125166    7 |   512  784  812  574  272   82  14   1
%e A125166    8 |   729 1296 1596 1386  846  354  96  15   1
%e A125166    9 |  1000 2025 2892 2982 2232 1200 450 111  16  1
%e A125166 ... refomatted and extended by _Wolfdieter Lang_, Mar 25 2025.
%t A125166 A125166[n_, k_] := A125166[n, k] = Switch[k, 0, (n + 1)^3, n, 1, _, A125166[n - 1, k - 1] + A125166[n - 1, k]];
%t A125166 Table[A125166[n, k], {n, 0, 10}, {k, 0, n}] (* _Paolo Xausa_, Apr 08 2025 *)
%o A125166 (SageMath)
%o A125166 y = polygen(QQ, 'y')
%o A125166 x = y.parent()[['x']].gen()
%o A125166 gf = (1 + 4*x + x^2)/((x - 1)^3*(y*x + x - 1))
%o A125166 [list(u) for u in list(gf.O(11))]  # _Peter Luschny_, Apr 02 2025
%Y A125166 Cf. A000578 (column 0), A000537 (column 1), A024166 (column 2), A101094 (column 3).
%Y A125166 Cf. A257448 (row sums).
%Y A125166 Cf. A008292, A028246, A097805, A125165, A050488, A382057.
%K A125166 nonn,easy,tabl
%O A125166 0,2
%A A125166 _Gary W. Adamson_, Nov 22 2006
%E A125166 a(27) corrected by _Georg Fischer_, Feb 18 2020