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Iterated differences of ludic numbers. Array read by antidiagonals, n >= 0, k >= 1: T(0,k) = A003309(k), T(n,k) = T(n-1,k+1)-T(n-1,k) for n > 0.
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%I #8 Oct 29 2023 20:31:23

%S 1,2,1,3,1,0,5,2,1,1,7,2,0,-1,-2,11,4,2,2,3,5,13,2,-2,-4,-6,-9,-14,17,

%T 4,2,4,8,14,23,37,23,6,2,0,-4,-12,-26,-49,-86,25,2,-4,-6,-6,-2,10,36,

%U 85,171,29,4,2,6,12,18,20,10,-26,-111,-282

%N Iterated differences of ludic numbers. Array read by antidiagonals, n >= 0, k >= 1: T(0,k) = A003309(k), T(n,k) = T(n-1,k+1)-T(n-1,k) for n > 0.

%H Winston de Greef, <a href="/A350004/b350004.txt">Table of n, a(n) for n = 0..11324</a> (150 antidiagonals)

%F T(n,k) = Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*A003309(k+j).

%e Array begins:

%e n\k| 1 2 3 4 5 6 7 8 9 10

%e ---+-----------------------------------------------

%e 0 | 1 2 3 5 7 11 13 17 23 25

%e 1 | 1 1 2 2 4 2 4 6 2 4

%e 2 | 0 1 0 2 -2 2 2 -4 2 4

%e 3 | 1 -1 2 -4 4 0 -6 6 2 -8

%e 4 | -2 3 -6 8 -4 -6 12 -4 -10 10

%e 5 | 5 -9 14 -12 -2 18 -16 -6 20 -8

%e 6 | -14 23 -26 10 20 -34 10 26 -28 2

%e 7 | 37 -49 36 10 -54 44 16 -54 30 8

%e 8 | -86 85 -26 -64 98 -28 -70 84 -22 -26

%e 9 | 171 -111 -38 162 -126 -42 154 -106 -4 64

%e 10 | -282 73 200 -288 84 196 -260 102 68 -142

%Y Cf. A003309 (row n = 0), A260723 (row n = 1).

%Y Cf. A095195 (iterated differences of primes), A350001 (iterated differences of lucky numbers).

%K sign,tabl

%O 0,2

%A _Pontus von Brömssen_, Dec 08 2021