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A103451 Triangular array T read by rows: T(n, 0) = T(n, n) = 1, T(n, k) = 0 for 0 < k < n.

%I A103451 #62 Jun 24 2025 14:54:48
%S A103451 1,1,1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0,0,0,1,1,0,0,0,0,0,1,1,0,0,0,0,0,
%T A103451 0,1,1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,1,1,0,
%U A103451 0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1
%N A103451 Triangular array T read by rows: T(n, 0) = T(n, n) = 1, T(n, k) = 0 for 0 < k < n.
%C A103451 Equals Pascal's triangle (A007318) where all elements > 1 are replaced with zero. Therefore it might be called "binomial skeleton".
%C A103451 Row sums are in A040000, antidiagonal sums are in A040001. When construed as a lower triangular matrix, the matrix inverse is A103452.
%H A103451 Michael De Vlieger, <a href="/A103451/b103451.txt">Rows n = 0..140 of triangle, flattened</a>
%H A103451 Carl M. Bender and Gerald V. Dunne, <a href="http://dx.doi.org/10.1063/1.527869">Polynomials and operator orderings</a>, J. Math. Phys. 29 (1988), 1727-1731.
%H A103451 Atli Fannar Franklín, <a href="https://arxiv.org/abs/2410.07467">Pattern avoidance enumerated by inversions</a>, arXiv:2410.07467 [math.CO], 2024. See pp. 2, 18.
%H A103451 Atli Fannar Franklín, Anders Claesson, Christian Bean, Henning Úlfarsson, and Jay Pantone, <a href="https://arxiv.org/abs/2406.16403">Restricted Permutations Enumerated by Inversions</a>, arXiv:2406.16403 [cs.DM], 2024. See p. 5.
%F A103451 a(n) = A097806(n-1) for n > 0. - _Philippe Deléham_, Oct 16 2007
%F A103451 T(n,k) = C(n,k-n) + C(n,-k) - C(0,n+k), 0 <= k <= n. - _Eric Werley_, Jul 01 2011
%F A103451 From _Stefano Spezia_, Jul 04 2024: (Start)
%F A103451 G.f.: (1 - x^2*y)/((1 - x)*(1 - x*y)).
%F A103451 E.g.f.: BesselI(0, 2*sqrt(x*y)) + exp(x) - 1. (End)
%e A103451 First few rows are:
%e A103451   1;
%e A103451   1, 1;
%e A103451   1, 0, 1;
%e A103451   1, 0, 0, 1;
%e A103451   1, 0, 0, 0, 1;
%e A103451   1, 0, 0, 0, 0, 1;
%e A103451   ...
%t A103451 Table[Boole[n == 0 || Mod[k, n] == 0], {n, 0, 14}, {k, 0, n}] (* or *)
%t A103451 Table[Binomial[n, k - n] + Binomial[n, -k] - Binomial[0, n + k], {n, 0, 14}, {k, 0, n}] // Flatten (* _Michael De Vlieger_, Jul 19 2016 *)
%o A103451 (Magma) r:=14; T:=ScalarMatrix(r, 1); for n in [1..r] do T[n, 1]:=1; end for; &cat[ [ T[n, k]: k in [1..n] ]: n in [1..r] ];
%o A103451 (Magma) /* As triangle */ [[Binomial(n, k-n)+Binomial(n, -k)-Binomial(0, n+k): k in [0..n]]: n in [0.. 15]]; // _Vincenzo Librandi_, Jul 20 2016
%o A103451 (PARI) for(n=0,15, for(k=0,n, print1(if(k==0||k==n, 1, 0), ", "))) \\ _G. C. Greubel_, Dec 08 2018
%o A103451 (Sage)
%o A103451 def A103451(n,k): return 1 if (k==0 or k==n) else 0
%o A103451 flatten([[A103451(n,k) for k in (0..n)] for n in (0..15)]) # _G. C. Greubel_, Feb 14 2021
%o A103451 (Python)
%o A103451 from math import isqrt, comb
%o A103451 def A103451(n):
%o A103451     if n==0: return 1
%o A103451     a = (m:=isqrt(k:=n+1<<1))-(k<=m*(m+1))
%o A103451     return int(not (n-comb(a+1,2))%a) # _Chai Wah Wu_, Jun 24 2025
%Y A103451 Cf. A007318, A040000, A040001, A097806, A103452.
%K A103451 easy,nonn,tabl
%O A103451 0,1
%A A103451 _Paul Barry_, Feb 06 2005
%E A103451 Edited by _Klaus Brockhaus_, Jan 26 2011