A131076 Row sums of triangular array T: T(j,1) = 1 for ((j-1) mod 8) < 4, else 0; T(j,k) = T(j-1,k-1) + T(j,k-1) for 2 <= k <= j.

1, 3, 7, 15, 26, 42, 64, 93, 139, 231, 463, 1092, 2744, 6840, 16384, 37383, 81295, 169119, 338239, 654192, 1232288, 2280864, 4194304, 7761375, 14635711, 28384383, 56768767, 116566080, 243472256, 511907712, 1073741824, 2232713343, 4585959679, 9292452351

Offset: 1

Author

Klaus Brockhaus, following a suggestion of Paul Curtz, Jun 14 2007

Keywords

• nonn
• easy

Comments

Sum of n-th row equals (n+1)-th term of main diagonal minus (n+1)-th term of first column: a(n) = A129961(n+1) - A131078(n+1).

Examples

			For first seven rows of T see A131074 or A129961.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (7,-20,30,-27,21,-24,30,-26,14,-4).

Crossrefs

Cf. A131074 (T read by rows), A129961 (main diagonal of T), A131075 (first subdiagonal of T), A131077 (antidiagonal sums of T). First through sixth column of T are in A131078, A131079, A131080, A131081, A131082, A131083 resp.

Programs

Magma

m:=32; M:=ZeroMatrix(IntegerRing(), m, m); for j:=1 to m do if (j-1) mod 8 lt 4 then M[j, 1]:=1; end if; end for; for k:=2 to m do for j:=k to m do M[j, k]:=M[j-1, k-1]+M[j, k-1]; end for; end for; [ &+[ M[j, k]: k in [1..j] ]: j in [1..m] ];

Mathematica

LinearRecurrence[{7,-20,30,-27,21,-24,30,-26,14,-4},{1,3,7,15,26,42,64,93,139,231},40] (* Harvey P. Dale, Jun 23 2025 *)

PARI

lista(m) = my(M=matrix(m, m)); for(j=1, m, M[j, 1]=if((j-1)%8<4, 1, 0)); for(k=2, m, for(j=k, m, M[j, k]=M[j-1, k-1]+M[j, k-1])); for(j=1, m, print1(sum(k=1, j, M[j, k]), ", "))

Formula

G.f.: x*(1-4*x+6*x^2-4*x^3-2*x^4+10*x^5-10*x^6+5*x^7-x^8)/((1-x)*(1-2*x)*(1+x^4)*(1-4*x+6*x^2-4*x^3+2*x^4)).