A047859 -id:A047859

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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).

+10
1

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..68.

EXAMPLE

First few rows of the triangle are:

2, 2;

3, 5, 3;

4, 9, 10, 4;

5, 14, 22, 17, 5;

6, 20, 40, 45, 26, 6

...

MAPLE

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

MATHEMATICA

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

CROSSREFS

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

KEYWORD

nonn,tabl

AUTHOR

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

EXTENSIONS

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

STATUS

approved

A210381

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

+10
1

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

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

REFERENCES

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

LINKS

Table of n, a(n) for n=0..65.

FORMULA

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

EXAMPLE

{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}

...

MATHEMATICA

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]

CROSSREFS

Cf. A074909, A255463, A026749, A047859, A027650

Cf. A099594.

KEYWORD

nonn,tabl

AUTHOR

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

STATUS

approved

A131415

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

+10
0

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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

LINKS

Table of n, a(n) for n=0..54.

FORMULA

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

EXAMPLE

First few rows of the triangle are:

3, 1;

6, 4, 1;

11, 10, 5, 1;

20, 21, 15, 6, 1;

37, 41, 36, 21, 7, 1;

...

CROSSREFS

Cf. A007318, A000012, A006127, A047859.

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson, Jul 08 2007

STATUS

approved

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