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A110814 Inverse of a triangle of pyramidal numbers.

%I A110814 #23 Sep 06 2024 05:06:18
%S A110814 1,-3,1,7,-4,1,-15,11,-5,1,31,-26,16,-6,1,-63,57,-42,22,-7,1,127,-120,
%T A110814 99,-64,29,-8,1,-255,247,-219,163,-93,37,-9,1,511,-502,466,-382,256,
%U A110814 -130,46,-10,1,-1023,1013,-968,848,-638,386,-176,56,-11,1,2047,-2036,1981,-1816,1486,-1024,562,-232,67,-12,1,-4095,4083
%N A110814 Inverse of a triangle of pyramidal numbers.
%C A110814 Inverse of A110813. Array factors as (1/(1+2x),x)*(1/(1+x),x/(1+x)). Row sums are (-2)^n. Diagonal sums are (-1)^n*A008466(n+2). Signed version of A104709.
%H A110814 G. C. Greubel, <a href="/A110814/b110814.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A110814 Number triangle T(n, k) = Sum_{j=0..n} (-2)^(n-j)*binomial(j, k)*(-1)^(j-k).
%F A110814 Riordan array (1/(1+3x+2x^2), x/(1+x)).
%F A110814 T(n,k) = -3*T(n-1,k) + T(n-1,k-1) - 2*T(n-2,k) + 2*T(n-2,k-1), T(0,0)=1, T(n,k)=0 if k < 0 or if k > n. - _Philippe Deléham_, Nov 30 2013
%e A110814 Rows begin
%e A110814     1;
%e A110814    -3,   1;
%e A110814     7,  -4,   1;
%e A110814   -15,  11,  -5,   1;
%e A110814    31, -26,  16,  -6,   1;
%p A110814 A110814_row := proc(n) add((-1)^k*add(binomial(n,n-i)*x^(n-k-1),i=0..k),k=0..n-1); coeffs(sort(%)) end; seq(print(A110814_row(n)),n=1..6); # _Peter Luschny_, Sep 29 2011
%t A110814 T[n_, k_] := Sum[(-2)^(n - j)*Binomial[j, k]*(-1)^(j - k), {j, 0, n}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _G. C. Greubel_, Oct 19 2017 *)
%o A110814 (PARI) for(n=0,10, for(k=0,n, print1(sum(j=0,n, (-2)^(n-j)*(-1)^(j-k)* binomial(j,k)), ", "))) \\ _G. C. Greubel_, Oct 19 2017
%K A110814 easy,sign,tabl
%O A110814 0,2
%A A110814 _Paul Barry_, Aug 05 2005