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A157901 Triangle read by rows: A000012 * A157898.

%I A157901 #14 Aug 30 2025 05:07:20
%S A157901 1,1,1,2,2,2,2,4,4,4,3,6,10,8,8,3,9,16,24,16,16,4,12,28,40,56,32,32,4,
%T A157901 16,40,80,96,128,64,64,5,20,60,120,216,224,288,128,128,5,25,80,200,
%U A157901 336,560,512,640,256,256,6,30,110,280,616,896,1408,1152,1408,512,512
%N A157901 Triangle read by rows: A000012 * A157898.
%C A157901 Multiplication of the lower triangular matrix A157898 from the left by A000012 means: these are partial column sums of A157898.
%H A157901 G. C. Greubel, <a href="/A157901/b157901.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A157901 T(n,k) = Sum_{j=0..n} A157898(j,k).
%e A157901 First few rows of the triangle, n>=0:
%e A157901   1;
%e A157901   1,  1;
%e A157901   2,  2,  2;
%e A157901   2,  4,  4,   4;
%e A157901   3,  6, 10,   8,   8;
%e A157901   3,  9, 16,  24,  16,  16;
%e A157901   4, 12, 28,  40,  56,  32,  32;
%e A157901   4, 16, 40,  80,  96, 128,  64,  64;
%e A157901   5, 20, 60, 120, 216, 224, 288, 128, 128;
%e A157901   5, 25, 80, 200, 336, 560, 512, 640, 256, 256;
%t A157901 t[n_, k_]:= t[n,k]= If[k==0 || k==n, 2^(n-1) +Boole[n==0]/2, 2*t[n-1,k-1] +2*t[n-1,k] - (2 -Boole[n==2])*t[n-2,k-1]]; (* t = A059576 *)
%t A157901 A157898[n_, k_]:= A157898[n,k]= Sum[(-1)^(n-j)*Binomial[n,j]*t[j,k], {j,k,n}];
%t A157901 A157901[n_, k_]:= A157901[n,k]= Sum[A157898[j+k,k], {j,0,n-k}];
%t A157901 Table[A157901[n,k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Aug 27 2025 *)
%o A157901 (Magma)
%o A157901 A011782:= func< n | n eq 0 select 1 else 2^(n-1) >;
%o A157901 function t(n, k) // t = A059576
%o A157901   if k eq 0 or k eq n then return A011782(n);
%o A157901   else return 2*t(n-1, k-1) + 2*t(n-1, k) - (2 - 0^(n-2))*t(n-2, k-1);
%o A157901   end if; return t;
%o A157901 end function;
%o A157901 A157898:= func< n, k | (&+[(-1)^(n-j)*Binomial(n, j)*t(j, k): j in [k..n]]) >;
%o A157901 A157071:= func< n,k | (&+[A157898(j+k,k): j in [0..n-k]]) >;
%o A157901 [A157071(n, k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Aug 27 2025
%o A157901 (SageMath)
%o A157901 @CachedFunction
%o A157901 def t(n, k): # t = A059576
%o A157901     if (k==0 or k==n): return bool(n==0)/2 + 2^(n-1) # A011782
%o A157901     else: return 2*t(n-1, k-1) + 2*t(n-1, k) - (2 - 0^(n-2))*t(n-2, k-1)
%o A157901 def A157898(n, k): return sum((-1)^(n+k-j)*binomial(n, j+k)*t(j+k, k) for j in range(n-k+1))
%o A157901 def A157071(n,k): return sum(A157898(j+k,k) for j in range(n-k+1))
%o A157901 print(flatten([[A157071(n,k) for k in range(n+1)] for n in range(10)])) # _G. C. Greubel_, Aug 27 2025
%Y A157901 Cf. A000012, A105635, A157898.
%Y A157901 Columns: A004526 (k=0), A002620 (k=1), A006584 (k=2), 4*A096338 (k=3), 8*A177747 (k=4), 16*A299337 (k=5), 32*A178440 (k=6).
%Y A157901 Sums include: A105635(n+1) (row), A166486(n+1) (alternating sign diagonal), A232801(n+1) (diagonal).
%K A157901 nonn,tabl,changed
%O A157901 0,4
%A A157901 _Gary W. Adamson_ & _Roger L. Bagula_, Mar 08 2009
%E A157901 Edited by the Associate Editors of the OEIS, Apr 10 2009
%E A157901 More terms from _G. C. Greubel_, Aug 27 2025