cp's OEIS Frontend

A124800 Let M be a diagonal matrix with A007442 on the diagonal and P = Pascal's triangle as an infinite lower triangular matrix. Now read the triangle P*M by rows.

%I A124800 #13 Aug 10 2025 10:33:06
%S A124800 2,2,1,2,2,1,2,3,3,-1,2,4,6,-4,3,2,5,10,-10,15,-9,2,6,15,-20,45,-54,
%T A124800 23,2,7,21,-35,105,-189,161,-53,2,8,28,-56,210,-504,644,-424,115,2,9,
%U A124800 36,-84,378,-1134,1932,-1908,1035,-237,2,10,45,-120,630,-2268,4830,-6360,5175,-2370,457
%N A124800 Let M be a diagonal matrix with A007442 on the diagonal and P = Pascal's triangle as an infinite lower triangular matrix. Now read the triangle P*M by rows.
%C A124800 Row sums = primes.
%C A124800 Right border = A007442, (2, 1, 1, -1, 3, -9...), = inverse binomial transform of the primes.
%F A124800 p(x,n) = Sum_{k=0..n} prime(k + 1) * binomial(n,k) * x^k * (1 - x)^(n - k); t(n,m) = coefficients(p(x,n)). - _Roger L. Bagula_ and _Gary W. Adamson_, Oct 01 2008
%e A124800 Row 5: sum = 11 = p5 since (2 + 4 + 6 - 4 + 3) = 11.
%e A124800 Triangle begins:
%e A124800   {2},
%e A124800   {2, 1},
%e A124800   {2, 2, 1},
%e A124800   {2, 3, 3, -1},
%e A124800   {2, 4, 6, -4, 3},
%e A124800   {2, 5, 10, -10, 15, -9},
%e A124800   {2, 6, 15, -20, 45, -54, 23},
%e A124800   {2, 7, 21, -35, 105, -189, 161, -53},
%e A124800   {2, 8, 28, -56, 210, -504, 644, -424, 115},
%e A124800   {2, 9, 36, -84, 378, -1134, 1932, -1908, 1035, -237},
%e A124800   {2, 10, 45, -120, 630, -2268, 4830, -6360, 5175, -2370, 457}
%e A124800   ...
%t A124800 p[x_, n_] = Sum[Prime[k + 1]*Binomial[n, k]*x^k*(1 - x)^(n - k), {k, 0, n}];
%t A124800 Table[CoefficientList[ExpandAll[p[x, n]], x], {n, 0, 10}] // Flatten (* _Roger L. Bagula_ and _Gary W. Adamson_, Oct 01 2008 *)
%Y A124800 Cf. A007442.
%K A124800 tabl,sign
%O A124800 1,1
%A A124800 _Gary W. Adamson_, Nov 07 2006
%E A124800 More terms from _Roger L. Bagula_ and _Gary W. Adamson_, Oct 01 2008