A034869 Right half of Pascal's triangle.

1, 1, 2, 1, 3, 1, 6, 4, 1, 10, 5, 1, 20, 15, 6, 1, 35, 21, 7, 1, 70, 56, 28, 8, 1, 126, 84, 36, 9, 1, 252, 210, 120, 45, 10, 1, 462, 330, 165, 55, 11, 1, 924, 792, 495, 220, 66, 12, 1, 1716, 1287, 715, 286, 78, 13, 1, 3432, 3003, 2002, 1001, 364, 91, 14, 1

Offset

0,3

Comments

From R. J. Mathar, May 13 2006: (Start)

Also flattened table of the expansion coefficients of x^n in Chebyshev Polynomials T_k(x) of the first kind:

x^n is 2^(1-n) multiplied by the sum of floor(1+n/2) terms using only terms T_k(x) with even k if n even, only terms T_k(x) with odd k if n is odd and halving the coefficient a(..) in front of any T_0(x):

x^0=2^(1-0) a(0)/2 T_0(x)

x^1=2^(1-1) a(1) T_1(x)

x^2=2^(1-2) [a(2)/2 T_0(x)+a(3) T_2(x)]

x^3=2^(1-3) [a(4) T_1(x)+a(5) T_3(x)]

x^4=2^(1-4) [a(6)/2 T_0(x)+a(7) T_2(x) +a(8) T_4(x)]

x^5=2^(1-5) [a(9) T_1(x)+a(10) T_3(x) +a(11) T_5(x)]

x^6=2^(1-6) [a(12)/2 T_0(x)+a(13) T_2(x) +a(14) T_4(x) +a(15) T_6(x)]

x^7=2^(1-7) [a(16) T_1(x)+a(17) T_3(x) +a(18) T_5(x) +a(19) T_7(x)]" (End)

T(n,k) = A034868(n,floor(n/2)-k), k = 0..floor(n/2). - Reinhard Zumkeller, Jul 27 2012

Rows are binomial(r-1,(2r+1-(-1)^r)\4 -n ) where r is the row and n is the term. Columns are binomial(2m+c-3,m-1) where c is the column and m is the term. - Anthony Browne, May 17 2016

Links

Reinhard Zumkeller, Rows n=0..150 of triangle, flattened

Formula

E.g.f. of column k: BesselI(2*k,2*x) + BesselI(2*k+1,2*x). - Mélika Tebni, Sep 05 2024

Example

The table starts:
  1
  1
  2 1
  3 1
  6 4 1
  ...

Maple

for n from 0 to 60 do for j from n mod 2 to n by 2 do print( binomial(n, (n-j)/2) ); od; od; # R. J. Mathar, May 13 2006
# Second program:
egf:= k-> BesselI(2*k, 2*x) + BesselI(2*k+1, 2*x):
A034869:= (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n):
seq(print(seq(A034869(n, k), k=0..iquo(n, 2))), n=0..14); # Mélika Tebni, Sep 05 2024

Mathematica

Table[Binomial[n, k], {n, 0, 14}, {k, Ceiling[n/2], n}] // Flatten (* Michael De Vlieger, May 19 2016 *)

Prog

(Haskell)
a034869 n k = a034869_tabf !! n !! k
a034869_row n = a034869_tabf !! n
a034869_tabf = [1] : f 0 [1] where
   f 0 us'@(_:us) = ys : f 1 ys where
                    ys = zipWith (+) us' (us ++ [0])
   f 1 vs@(v:_) = ys : f 0 ys where
                  ys = zipWith (+) (vs ++ [0]) ([v] ++ vs)
-- Reinhard Zumkeller, improved Dec 21 2015, Jul 27 2012
(PARI) for(n=0, 14, for(k=ceil(n/2), n, print1(binomial(n, k), ", "); ); print(); ) \\ Indranil Ghosh, Mar 31 2017
(Python)
import math
from sympy import binomial
for n in range(15):
    print([binomial(n, k) for k in range(math.ceil(n/2), n + 1)]) # Indranil Ghosh, Mar 31 2017

Crossrefs

Cf. A007318, A008619 (row lengths).

Cf. A110654.

Cf. A034868 (left half), A014413, A014462, A027306 (row sums).

Columns k=0-1-2-3-4 give: A001405, A037955, A037956, A037957, A037958.

Sequence in context: A336105 A282601 A363273 * A205858 A340793 A053222

Adjacent sequences: A034866 A034867 A034868 * A034870 A034871 A034872

Keyword

nonn,tabf,easy

Author

N. J. A. Sloane.

Extensions

Keyword fixed and example added by Franklin T. Adams-Watters, May 27 2010

Status

approved