cp's OEIS Frontend

A034869 Right half of Pascal's triangle.

%I A034869 #50 Jul 08 2025 21:28:03
%S A034869 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,
%T A034869 9,1,252,210,120,45,10,1,462,330,165,55,11,1,924,792,495,220,66,12,1,
%U A034869 1716,1287,715,286,78,13,1,3432,3003,2002,1001,364,91,14,1
%N A034869 Right half of Pascal's triangle.
%C A034869 From _R. J. Mathar_, May 13 2006: (Start)
%C A034869 Also flattened table of the expansion coefficients of x^n in Chebyshev Polynomials T_k(x) of the first kind:
%C A034869 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):
%C A034869 x^0=2^(1-0) a(0)/2 T_0(x)
%C A034869 x^1=2^(1-1) a(1) T_1(x)
%C A034869 x^2=2^(1-2) [a(2)/2 T_0(x)+a(3) T_2(x)]
%C A034869 x^3=2^(1-3) [a(4) T_1(x)+a(5) T_3(x)]
%C A034869 x^4=2^(1-4) [a(6)/2 T_0(x)+a(7) T_2(x) +a(8) T_4(x)]
%C A034869 x^5=2^(1-5) [a(9) T_1(x)+a(10) T_3(x) +a(11) T_5(x)]
%C A034869 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)]
%C A034869 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)
%C A034869 T(n,k) = A034868(n,floor(n/2)-k), k = 0..floor(n/2). - _Reinhard Zumkeller_, Jul 27 2012
%C A034869 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
%H A034869 Reinhard Zumkeller, <a href="/A034869/b034869.txt">Rows n=0..150 of triangle, flattened</a>
%F A034869 E.g.f. of column k: BesselI(2*k,2*x) + BesselI(2*k+1,2*x). - _Mélika Tebni_, Sep 05 2024
%e A034869 The table starts:
%e A034869   1
%e A034869   1
%e A034869   2 1
%e A034869   3 1
%e A034869   6 4 1
%e A034869   ...
%p A034869 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
%p A034869 # Second program:
%p A034869 egf:= k-> BesselI(2*k, 2*x) + BesselI(2*k+1, 2*x):
%p A034869 A034869:= (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n):
%p A034869 seq(print(seq(A034869(n, k), k=0..iquo(n, 2))), n=0..14); # _Mélika Tebni_, Sep 05 2024
%t A034869 Table[Binomial[n, k], {n, 0, 14}, {k, Ceiling[n/2], n}] // Flatten (* _Michael De Vlieger_, May 19 2016 *)
%o A034869 (Haskell)
%o A034869 a034869 n k = a034869_tabf !! n !! k
%o A034869 a034869_row n = a034869_tabf !! n
%o A034869 a034869_tabf = [1] : f 0 [1] where
%o A034869    f 0 us'@(_:us) = ys : f 1 ys where
%o A034869                     ys = zipWith (+) us' (us ++ [0])
%o A034869    f 1 vs@(v:_) = ys : f 0 ys where
%o A034869                   ys = zipWith (+) (vs ++ [0]) ([v] ++ vs)
%o A034869 -- _Reinhard Zumkeller_, improved Dec 21 2015, Jul 27 2012
%o A034869 (PARI) for(n=0, 14, for(k=ceil(n/2), n, print1(binomial(n, k),", ");); print();) \\ _Indranil Ghosh_, Mar 31 2017
%o A034869 (Python)
%o A034869 import math
%o A034869 from sympy import binomial
%o A034869 for n in range(15):
%o A034869     print([binomial(n, k) for k in range(math.ceil(n/2), n + 1)]) # _Indranil Ghosh_, Mar 31 2017
%Y A034869 Cf. A007318, A008619 (row lengths).
%Y A034869 Cf. A110654.
%Y A034869 Cf. A034868 (left half), A014413, A014462, A027306 (row sums).
%Y A034869 Columns k=0-1-2-3-4 give: A001405, A037955, A037956, A037957, A037958.
%K A034869 nonn,tabf,easy
%O A034869 0,3
%A A034869 _N. J. A. Sloane_
%E A034869 Keyword fixed and example added by _Franklin T. Adams-Watters_, May 27 2010