A165312 -id:A165312

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A165253

Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

+10
14

1, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 5, 1, 0, 1, 10, 15, 7, 1, 0, 1, 15, 35, 28, 9, 1, 0, 1, 21, 70, 84, 45, 11, 1, 0, 1, 28, 126, 210, 165, 66, 13, 1, 0, 1, 36, 210, 462, 495, 286, 91, 15, 1, 0, 1, 45, 330, 924, 1287, 1001, 455, 120, 17, 1, 0, 1, 55, 495, 1716, 3003, 3003, 1820, 680

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

OFFSET

0,8

COMMENTS

Mirror image of triangle in A121314.

LINKS

Indranil Ghosh, Rows 0..125 of triangle, flattened

FORMULA

T(0,0)=1, T(n,k) = binomial(n-1+k,2k) for n >= 1.

Sum {k=0..n} T(n,k)*x^k = A000012(n), A001519(n), A001835(n), A004253(n), A001653(n), A049685(n-1), A070997(n-1), A070998(n-1), A072256(n), A078922(n), A077417(n-1), A085260(n), A001570(n) for x = 0,1,2,3,4,5,6,7,8,9,10,11,12 respectively.

Sum_{k=0..n} T(n,k)*x^(n-k) = A000007(n), A001519(n), A047849(n), A165310(n), A165311(n), A165312(n), A165314(n), A165322(n), A165323(n), A165324(n) for x= 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively. - Philippe Deléham, Sep 26 2009

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k), T(0,0)=T(1,0)=1, T(1,1)=0. - Philippe Deléham, Feb 18 2012

G.f.: (1-x-y*x)/((1-x)^2-y*x). - Philippe Deléham, Feb 19 2012

EXAMPLE

Triangle begins:

1, 0;

1, 1, 0;

1, 3, 1, 0;

1, 6, 5, 1, 0;

1, 10, 15, 7, 1, 0;

1, 15, 35, 28, 9, 1, 0;

1, 21, 70, 84, 45, 11, 1, 0;

1, 28, 126, 210, 165, 66, 13, 1, 0;

1, 36, 210, 462, 495, 286, 91, 15, 1, 0,

1, 45, 330, 924, 1287, 1001, 455, 120, 17, 1, 0;

MATHEMATICA

m = 13;

(* DELTA is defined in A084938 *)

DELTA[Join[{1, 0, 1}, Table[0, {m}]], Join[{0, 1}, Table[0, {m}]], m] // Flatten (* Jean-François Alcover, Feb 19 2020 *)

CROSSREFS

Cf. A054142, A085478, A121314.

KEYWORD

nonn,tabl

AUTHOR

Philippe Deléham, Sep 10 2009

STATUS

approved

A121314

Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

+10
8

1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 5, 6, 1, 0, 1, 7, 15, 10, 1, 0, 1, 9, 28, 35, 15, 1, 0, 1, 11, 45, 84, 70, 21, 1, 0, 1, 13, 66, 165, 210, 126, 28, 1, 0, 1, 15, 91, 286, 495, 462, 210, 36, 1

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

OFFSET

0,9

COMMENTS

A054142 with first diagonal 1, 0, 0, 0, 0, 0, 0, 0, ...

Mirror image of triangle in A165253.

LINKS

Alois P. Heinz, Rows n = 0..140, flattened

F. Yano and H. Yoshida, Some set partition statistics in non-crossing partitions and generating functions, Discr. Math., 307 (2007), 3147-3160.

FORMULA

T(0,0)=1; T(n,0)=0 for n > 0; T(n+1,k+1) = binomial(2*n-k,k)for n >= 0 and k >= 0.

Sum_{k=0..n} T(n,k)*x^k = A001519(n), A047849(n), A165310(n), A165311(n), A165312(n), A165314(n), A165322(n), A165323(n), A165324(n) for x = 1,2,3,4,5,6,7,8,9 respectively.

Sum_{k=0..n} 2^k*T(n,k) = (4^n+2)/3.

Sum_{k=0..n} 2^(n-k)*T(n,k) = A001835(n).

Sum_{k=0..n} 3^k*4^(n-k)*T(n,k) = A054879(n). - Philippe Deléham, Aug 26 2006

Sum_{k=0..n} T(n,k)*(-1)^k*2^(3n-2k) = A143126(n). - Philippe Deléham, Oct 31 2008

Sum_{k=0..n} T(n,k)*(-1)^k*3^(n-k) = A138340(n)/4^n. - Philippe Deléham, Nov 01 2008

G.f.: (1-(y+1)*x)/(1-(2y+1)*x+y^2*x^2). - Philippe Deléham, Nov 01 2011

T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-2), T(0,0) = T(1,1) = 1, T(1,0) = 0. - Philippe Deléham, Feb 19 2012

EXAMPLE

Triangle begins

0, 1;

0, 1, 1;

0, 1, 3, 1;

0, 1, 5, 6, 1;

0, 1, 7, 15, 10, 1;

0, 1, 9, 28, 35, 15, 1;

0, 1, 11, 45, 84, 70, 21, 1;