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A125790 Rectangular table where column k equals row sums of matrix power A078121^k, read by antidiagonals.

%I A125790 #27 May 29 2025 22:03:25
%S A125790 1,1,1,1,2,1,1,4,3,1,1,10,9,4,1,1,36,35,16,5,1,1,202,201,84,25,6,1,1,
%T A125790 1828,1827,656,165,36,7,1,1,27338,27337,8148,1625,286,49,8,1,1,692004,
%U A125790 692003,167568,25509,3396,455,64,9,1,1,30251722,30251721,5866452,664665,64350,6321,680,81,10,1
%N A125790 Rectangular table where column k equals row sums of matrix power A078121^k, read by antidiagonals.
%C A125790 Determinant of n X n upper left submatrix is 2^[n(n-1)(n-2)/6] (see A125791). Related to partitions of numbers into powers of 2 (see A078121). Triangle A078121 shifts left one column under matrix square.
%H A125790 G. Blom and C.-E. Froeberg, <a href="/A002575/a002575.pdf">Om myntvaexling (On money-changing) [Swedish]</a>, Nordisk Matematisk Tidskrift, 10 (1962), 55-69, 103. [Annotated scanned copy] See Table 5.
%F A125790 T(n,k) = T(n,k-1) + T(n-1,2*k) for n>0, k>0, with T(0,n)=T(n,0)=1 for n>=0.
%F A125790 Conjecture: g.f. for n-th row is (Sum_{i=0..n-1} x^i Sum_{j=0..i} binomial(n+1,j)*T(n,i-j)*(-1)^j)/(1-x)^(n+1) for n > 0. - _Mikhail Kurkov_, May 03 2025
%e A125790 Recurrence T(n,k) = T(n,k-1) + T(n-1,2*k) is illustrated by:
%e A125790   T(4,3) = T(4,2) + T(3,6) = 201 + 455 = 656;
%e A125790   T(5,3) = T(5,2) + T(4,6) = 1827 + 6321 = 8148;
%e A125790   T(6,3) = T(6,2) + T(5,6) = 27337 + 140231 = 167568.
%e A125790 Rows of this table begin:
%e A125790   1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...;
%e A125790   1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, ...;
%e A125790   1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, ...;
%e A125790   1, 10, 35, 84, 165, 286, 455, 680, 969, 1330, 1771, 2300, ...;
%e A125790   1, 36, 201, 656, 1625, 3396, 6321, 10816, 17361, 26500, 38841, ...;
%e A125790   1, 202, 1827, 8148, 25509, 64350, 140231, 274856, 497097, ...;
%e A125790   1, 1828, 27337, 167568, 664665, 2026564, 5174449, 11622976, ...;
%e A125790   1, 27338, 692003, 5866452, 29559717, 109082974, 326603719, ...;
%e A125790   1, 692004, 30251721, 356855440, 2290267225, 10243585092, ...; ...
%e A125790 Triangle A078121 begins:
%e A125790   1;
%e A125790   1,   1;
%e A125790   1,   2,   1;
%e A125790   1,   4,   4,   1;
%e A125790   1,  10,  16,   8,   1;
%e A125790   1,  36,  84,  64,  16,  1;
%e A125790   1, 202, 656, 680, 256, 32, 1; ...
%e A125790 where row sums form column 1 of this table A125790,
%e A125790 and column k of A078121 equals column 2^k-1 of this table A125790.
%e A125790 Matrix cube A078121^3 begins:
%e A125790      1;
%e A125790      3,    1;
%e A125790      9,    6,    1;
%e A125790     35,   36,   12,   1;
%e A125790    201,  286,  144,  24,  1;
%e A125790   1827, 3396, 2300, 576, 48, 1; ...
%e A125790 where row sums form column 3 of this table A125790,
%e A125790 and column 0 of A078121^3 forms column 2 of this table A125790.
%t A125790 T[n_, k_] := T[n, k] = T[n, k-1] + T[n-1, 2*k]; T[0, _] = T[_, 0] = 1; Table[T[n-k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jun 15 2015 *)
%o A125790 (PARI) {T(n,k,p=0,q=2)=local(A=Mat(1), B); if(n<p||p<0, 0, for(m=1, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i||j==1, B[i, j]=1, B[i, j]=(A^q)[i-1, j-1]); )); A=B); return((A^(k+1))[n+1, p+1]))}
%o A125790 for(n=0,10,for(k=0,10,print1(T(n,k),", ")); print(""))
%Y A125790 Cf. A078121; A002577; A125791; columns: A002577, A125792, A125793, A125794, A125795, A125796; diagonals: A125797, A125798; A125799 (antidiagonal sums); related table: A125800 (q=3).
%K A125790 nonn,tabl
%O A125790 0,5
%A A125790 _Paul D. Hanna_, Dec 10 2006, corrected Dec 12 2006