login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A054143 Triangular array T given by T(n,k) = Sum_{0<=j<=i-n+k, n-k<=i<=n} C(i,j). 11
1, 1, 3, 1, 4, 7, 1, 5, 11, 15, 1, 6, 16, 26, 31, 1, 7, 22, 42, 57, 63, 1, 8, 29, 64, 99, 120, 127, 1, 9, 37, 93, 163, 219, 247, 255, 1, 10, 46, 130, 256, 382, 466, 502, 511, 1, 11, 56, 176, 386, 638, 848, 968, 1013, 1023, 1, 12, 67, 232, 562, 1024, 1486, 1816, 1981, 2036, 2047 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Row sums given by A001787.

T(n, n) = -1 + 2^(n+1).

T(2n, n) = 4^n.

T(2n+1, n) = A000346(n).

T(2n-1, n) = A032443(n).

A054143 is the fission of the polynomial sequence ((x+1^n) by the polynomial sequence (q(n,x)) given by q(n,x) = x^n + x^(n-1) + ... + x + 1. See A193842 for the definition of fission. - Clark Kimberling, Aug 07 2011

LINKS

G. C. Greubel, Rows n = 0..100 of triangle, flattened

FORMULA

T(n,k) = Sum_{0<=j<=i-n+k, n-k<=i<=n} binomial(i,j).

T(n,k) = T(n-1,k) + 3*T(n-1,k-1) - 2*T(n-2,k-1) - 2*T(n-2,k-2), T(0,0)=1, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Nov 30 2013

EXAMPLE

First six rows:

  1;

  1,  3;

  1,  4,  7;

  1,  5, 11, 15;

  1,  6, 16, 26, 31;

  1,  7, 22, 42, 57, 63;

MAPLE

A054143_row := proc(n) add(add(binomial(n, n-i)*x^(k+1), i=0..k), k=0..n-1); coeffs(sort(%)) end; seq(print(A054143_row(n)), n=1..6); # Peter Luschny, Sep 29 2011

MATHEMATICA

(* First program *)

z=10;

p[n_, x_]:=(x+1)^n;

q[0, x_]:=1; q[n_, x_]:=x*q[n-1, x]+1;

p1[n_, k_]:=Coefficient[p[n, x], x^k]; p1[n_, 0]:=p[n, x]/.x->0;

d[n_, x_]:=Sum[p1[n, k]*q[n-1-k, x], {k, 0, n-1}]

h[n_]:=CoefficientList[d[n, x], {x}]

TableForm[Table[Reverse[h[n]], {n, 0, z}]]

Flatten[Table[Reverse[h[n]], {n, -1, z}]] (* A054143 *)

TableForm[Table[h[n], {n, 0, z}]]

Flatten[Table[h[n], {n, -1, z}]] (* A104709 *)

(* Second program *)

Table[Sum[Binomial[i, j], {i, n-k, n}, {j, 0, i-n+k}], {n, 0, 12}, {k, 0, n}]// Flatten (* G. C. Greubel, Aug 01 2019 *)

PROG

(PARI) T(n, k) = sum(i=n-k, n, sum(j=0, i-n+k, binomial(i, j)));

for(n=0, 12, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Aug 01 2019

(MAGMA)

T:= func< n, k | (&+[ (&+[ Binomial(i, j): j in [0..i-n+k]]): i in [n-k..n]]) >;

[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 01 2019

(Sage)

def T(n, k): return sum(sum( binomial(i, j) for j in (0..i-n+k)) for i in (n-k..n))

[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Aug 01 2019

(GAP) Flat(List([0..12], n-> List([0..n], k-> Sum([n-k..n], i-> Sum([0..i-n+k], j-> Binomial(i, j) ))))); # G. C. Greubel, Aug 01 2019

CROSSREFS

Cf. A000346, A001787, A032443.

Diagonal sums give A005672. - Paul Barry, Feb 07 2003

Sequence in context: A213224 A210218 A086273 * A104746 A208339 A328463

Adjacent sequences:  A054140 A054141 A054142 * A054144 A054145 A054146

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 18 2000

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 16 06:18 EST 2019. Contains 330016 sequences. (Running on oeis4.)