A121775 -id:A121775 - cp's OEIS frontend

A121776 Antidiagonal sums of triangle A121775.

1, 1, 3, 6, 10, 16, 23, 36, 52, 76, 118, 181, 271, 427, 675, 1057, 1686, 2705, 4318, 6923, 11142, 17966, 28999, 46793, 75522, 122018, 197254, 318907, 515616, 833796, 1348542, 2181304, 3528487, 5707947, 9234075, 14938991, 24169111, 39103056, 63265607, 102359886

Offset: 0

Paul D. Hanna, Aug 23 2006

nonn

Cf. A000010, A121775.

a[0] = 1; a[n_] := Sum[DivisorSum[n-k, EulerPhi[(n-k)/#] * Binomial[#, k] &], {k, 0, Floor[n/2]}]; Array[a, 50, 0] (* Amiram Eldar, Aug 15 2023 *)

a(n)=if(n==0,1,sum(k=0,n\2,sumdiv(n-k,d,eulerphi((n-k)/d)*binomial(d,k))))

a(n) = Sum_{k=0..[n/2]} Sum_{d|(n-k)} phi((n-k)/d)*C(d,k) for n>0, with a(0)=1.

A130301 Triangle read by rows: A130296 * A007318, as infinite lower triangular matrices.

Original entry on oeis.org

1, 3, 1, 5, 3, 1, 7, 6, 4, 1, 9, 10, 10, 5, 1, 11, 15, 20, 15, 6, 1, 13, 21, 35, 35, 21, 7, 1, 15, 28, 56, 70, 56, 28, 8, 1, 17, 36, 84, 126, 126, 84, 36, 9, 1, 19, 45, 120, 210, 252, 210, 120, 45, 10, 1, 21, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1

Offset: 1

Gary W. Adamson, May 20 2007

Row sums = A083706: (1, 4, 9, 18, 35, 68, ...).
A130300 = A007318 * A130296.
The lower triangular matrix A130296 is equal to the restriction of the square array A051340 to its lower left triangular part. So this is also equal to (A051340) * A007318, where (A051340) is the lower triangular part of A051340, i.e., A051340[i,j] replaced by zero for j > i: see  Mathar's Maple code. - M. F. Hasler, Aug 15 2015

			First few rows of the triangle:
   1;
   3,  1;
   5,  3,  1;
   7,  6,  4,  1;
   9, 10, 10,  5,  1;
  11, 15, 20, 15,  6,  1;
  13, 21, 35, 35, 21,  7,  1;
  ...

Cf. A083706, A130296, A130300.

A130301 := proc(n, k)
    add( A051340(n, i)*binomial(i, k), i=k..n);
end proc: # R. J. Mathar, Jul 16 2015

A130301[m,n] = A121775[m,n] for n >= m/2. A130301[m,1] = 2m-1, A130301[m,2] = A000217[m-1], A130301[m,m] = 1, A130301[m,m-1] = m for m>2. - M. F. Hasler, Aug 15 2015

Corrected (missing a(15)=1 inserted) by M. F. Hasler, Aug 15 2015

a(26) = 27 corrected and more terms from Georg Fischer, May 29 2023

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A121776 Antidiagonal sums of triangle A121775.

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A130301 Triangle read by rows: A130296 * A007318, as infinite lower triangular matrices.

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