A128621 -id:A128621 - cp's OEIS frontend

A128623 Triangle read by rows: A128621 * A000012 as infinite lower triangular matrices.

1, 2, 2, 6, 3, 3, 8, 8, 4, 4, 15, 10, 10, 5, 5, 18, 18, 12, 12, 6, 6, 28, 21, 21, 14, 14, 7, 7, 32, 32, 24, 24, 16, 16, 8, 8, 45, 36, 36, 27, 27, 18, 18, 9, 9, 50, 50, 40, 40, 30, 30, 20, 20, 10, 10, 66, 55, 55, 44, 44, 33, 33, 22, 22, 11, 11, 72, 72, 60, 60, 48, 48, 36, 36, 24, 24, 12, 12, 91, 78, 78, 65, 65, 52, 52, 39, 39, 26, 26, 13, 13

Offset: 1

Gary W. Adamson, Mar 14 2007

			First few rows of the triangle are:
   1;
   2,  2;
   6,  3,  3;
   8,  8,  4,  4;
  15, 10, 10,  5,  5;
  18, 18, 12, 12,  6, 6;
  28, 21, 21, 14, 14, 7, 7;
  ...

G. C. Greubel, Rows n = 1..100 of the triangle, flattened

Cf. A000027, A093005, A093353, A115514, A128621, A245524.

Cf. A128624 (row sums).

[n*Floor((n-k+2)/2): k in [1..n], n in [1..15]]; // G. C. Greubel, Mar 13 2024

Table[n*Floor[(n-k+2)/2], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Mar 13 2024 *)

flatten([[n*((n-k+2)//2) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Mar 13 2024

Sum_{k=1..n} T(n, k) = A128624(n) (row sums).
T(n,k) = n*(1+floor((n-k)/2)), 1 <= k <= n. - R. J. Mathar, Jun 27 2012
From G. C. Greubel, Mar 13 2024: (Start)
T(n, k) = n*A115514(n, k).
T(n, k) = Sum_{j=k..n} A128621(n, j).
T(n, 1) = A093005(n).
T(n, 2) = A093353(n-1), n >= 2.
T(n, n) = A000027(n).
T(2*n-1, n) = A245524(n).
Sum_{k=1..n} (-1)^k*T(n, k) = (1/2)*(1-(-1)^n)*A000384(floor((n+1)/2)). (End)

a(41) = 27 inserted and more terms from Georg Fischer, Jun 05 2023

A128624 Row sums of A128623.

Original entry on oeis.org

1, 4, 12, 24, 45, 72, 112, 160, 225, 300, 396, 504, 637, 784, 960, 1152, 1377, 1620, 1900, 2200, 2541, 2904, 3312, 3744, 4225, 4732, 5292, 5880, 6525, 7200, 7936, 8704, 9537, 10404, 11340, 12312, 13357, 14440, 15600, 16800, 18081, 19404, 20812, 22264, 23805

Offset: 1

Gary W. Adamson, Mar 14 2007

Also the number of (w,x,y) with all terms in {0,...,n-1} and w <= R <= x, where R = max(w,x,y)-min(w,x,y), see A212959. - Clark Kimberling, Jun 10 2012

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A128621, A128623, A212959.

Cf. A094728 (diagonal row sums).

[n*((n+1)^2-1+(n mod 2))/4: n in [1..50]]; // G. C. Greubel, Mar 12 2024

Table[n*(n^2 +2*n +Mod[n,2])/4, {n,50}] (* G. C. Greubel, Mar 12 2024 *)

Vec(x*(1+2*x+3*x^2)/((1-x)^4*(1+x)^2) + O(x^100)) \\ Colin Barker, Jan 31 2016

[n*((n+1)^2-1+(n%2))//4 for n in range(1,51)] # G. C. Greubel, Mar 12 2024

G.f.: x*(1+2*x+3*x^2) / ((1+x)^2*(1-x)^4). - R. J. Mathar, Jun 27 2012
From Colin Barker, Jan 31 2016: (Start)
a(n) = n*(2*n^2 + 4*n + 1 - (-1)^n)/8.
a(n) = n^2*(n + 2)/4 for n even.
a(n) = n*(n^2 + 2*n + 1)/4 for n odd. (End)
From G. C. Greubel, Mar 12 2024: (Start)
a(n) = Sum_{k=0..floor((n-1)/2)} A094728(n, k).
E.g.f.: (1/8)*x*(exp(-x) + (7 + 10*x + 2*x^2)*exp(x)). (End)

Incorrect formula removed by R. J. Mathar, Jun 27 2012

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A128623 Triangle read by rows: A128621 * A000012 as infinite lower triangular matrices.

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