A047859 -id:A047859 - cp's OEIS frontend

A124727 Triangle read by rows: T(n,k)=k*binomial(n-1,k-1)+binomial(n-1,k) (1<=k<=n).

1, 2, 2, 3, 5, 3, 4, 9, 10, 4, 5, 14, 22, 17, 5, 6, 20, 40, 45, 26, 6, 7, 27, 65, 95, 81, 37, 7, 8, 35, 98, 175, 196, 133, 50, 8, 9, 44, 140, 294, 406, 364, 204, 65, 9, 10, 54, 192, 462, 756, 840, 624, 297, 82, 10, 11, 65, 255, 690, 1302, 1722, 1590, 1005, 415, 101, 11, 12, 77

Offset: 1

Gary W. Adamson & Roger L. Bagula, Nov 05 2006

Triangle is P*M, where P is Pascal's triangle as an infinite lower triangular matrix and M is the infinite bidiagonal matrix with (1,2,3...) in the main diagonal and (1,1,1...) in the subdiagonal.

			First few rows of the triangle are:
1;
2, 2;
3, 5, 3;
4, 9, 10, 4;
5, 14, 22, 17, 5;
6, 20, 40, 45, 26, 6
...

Row sums = A047859: (1, 4, 11, 27, 143, 319...) A124726 is generated in an analogous manner by taking M*P instead of P*M.

T:=(n,k)->k*binomial(n-1,k-1)+binomial(n-1,k): for n from 1 to 12 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form

Flatten[Table[k Binomial[n-1,k-1]+Binomial[n-1,k],{n,20},{k,n}]] (* Harvey P. Dale, Jan 28 2012 *)

Edited by N. J. A. Sloane, Nov 24 2006

A210381 Triangle by rows, derived from the beheaded Pascal's triangle, A074909.

Original entry on oeis.org

1, 0, 2, 0, 1, 3, 0, 1, 3, 4, 0, 1, 4, 6, 5, 0, 1, 5, 10, 10, 6, 0, 1, 6, 15, 20, 15, 7, 0, 1, 7, 21, 35, 35, 21, 8, 0, 1, 8, 28, 56, 70, 56, 28, 9, 0, 1, 9, 36, 84, 126, 126, 84, 36, 10, 0, 1, 10, 45, 120, 210, 252, 210, 120, 45, 11

Offset: 0

Roger L. Bagula and Gary W. Adamson, Mar 20 2012

Row sums of the triangle = 2^n.
Let the triangle = an infinite lower triangular matrix, M.  Then M * The Bernoulli numbers, A027641/A027642  as a vector V = [1, -1, 0, 0, 0,...].  M * the Bernoulli sequence variant starting [1, 1/2, 1/6,...] = [1, 1, 1,...].  M * 2^n: [1, 2, 4, 8,...] = A027649.  M * 3^n = A255463; while M * [1, 2, 3,...] = A047859, and M * A027649 = A027650.
Row sums of powers of the triangle generate the Poly-Bernoulli number sequences shown in the array of A099594. - Gary W. Adamson, Mar 21 2012
Triangle T(n,k) given by (0, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 25 2012

			{1},
{0, 2},
{0, 1, 3},
{0, 1, 3, 4},
{0, 1, 4, 6, 5},
{0, 1, 5, 10, 10, 6},
{0, 1, 6, 15, 20, 15, 7},
{0, 1, 7, 21, 35, 35, 21, 8},
{0, 1, 8, 28, 56, 70, 56, 28, 9},
{0, 1, 9, 36, 84, 126, 126, 84, 36, 10},
{0, 1, 10, 45, 120, 210, 252, 210, 120, 45, 11}
...

Konrad Knopp, Elements of the Theory of Functions, Dover, 1952,pp 117-118.

Cf. A074909, A255463, A026749, A047859, A027650

Cf. A099594.

t2[n_, m_] = If[m - 1 <= n, Binomial[n, m - 1], 0];
O2 = Table[Table[If[n == m, t2[n, m] + 1, t2[n, m]], {m, 0, n}], {n, 0, 10}];
Flatten[O2]

Partial differences of the beheaded Pascal's triangle A074909 starting from the top, by columns.
G.f.: (1-x)/(1-x-2*y*x+y*x^2+y^2*x^2). - Philippe Deléham, Mar 25 2012
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1) - T(n-2,k-2), T(0,0) = T(2,1) = 1, T(1,0) = T(2,0) = 0, T(1,1) = 2, T(2,2) = 3 and T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Mar 25 2012

A131415 (A007318 * A000012) + (A000012 * A007318) - A007318.

Original entry on oeis.org

1, 3, 1, 6, 4, 1, 11, 10, 5, 1, 20, 21, 15, 6, 1, 37, 41, 36, 21, 7, 1, 70, 78, 77, 57, 28, 8, 1, 135, 148, 155, 134, 85, 36, 9, 1, 264, 283, 303, 289, 219, 121, 45, 10, 1, 521, 547, 586, 592, 508, 340, 166, 55, 11, 1

Offset: 0

Gary W. Adamson, Jul 08 2007

Left column = A006127: (1, 3, 6, 11, 20, 37,...). Row sums = A047859: (1, 4, 11, 27, 63,...).

			First few rows of the triangle are:
1;
3, 1;
6, 4, 1;
11, 10, 5, 1;
20, 21, 15, 6, 1;
37, 41, 36, 21, 7, 1;
...

Cf. A007318, A000012, A006127, A047859.

(A007318 * A000012) + (A000012 * A007318) - A007318 as infinite lower triangular matrices.

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A124727 Triangle read by rows: T(n,k)=k*binomial(n-1,k-1)+binomial(n-1,k) (1<=k<=n).

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A210381 Triangle by rows, derived from the beheaded Pascal's triangle, A074909.

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A131415 (A007318 * A000012) + (A000012 * A007318) - A007318.

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