Original entry on oeis.org
0, 1, 0, 0, 2, 0, -1, 2, 3, 0, -3, 4, 5, 4, 0, -9, 14, 15, 9, 5, 0, -33, 68, 65, 34, 14, 6, 0, -153, 404, 359, 174, 63, 20, 7, 0, -873, 2804, 2375, 1098, 371, 104, 27, 8, 0, -5913, 22244, 18215, 8154, 2639, 692, 159, 35, 9, 0, -46233, 198644, 158615, 69354, 21791, 5480, 1179, 230, 44, 10, 0
Offset: 0
Triangle begins:
0;
1, 0;
0, 2, 0;
-1, 2, 3, 0;
-3, 4, 5, 4, 0;
-9, 14, 15, 9, 5, 0;
-33, 68, 65, 34, 14, 6, 0;
-153, 404, 359, 174, 63, 20, 7, 0;
-873, 2804, 2375, 1098, 371, 104, 27, 8, 0;
-5913, 22244, 18215, 8154, 2639, 692, 159, 35, 9, 0;
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m=12;
M= Table[If[k>n-1, 0, If[k==n-1, n, -1]], {n,0,m+1}, {k,0,m+1}];
T:= T= Sum[MatrixPower[M, j]/j, {j,m+1}];
Table[T[[n+1, k+1]], {n,0,m}, {k,0,n}]//Flatten (* G. C. Greubel, Sep 06 2022 *)
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{T(n,k)=local(M=matrix(n+4,n+4,r,c,if(r>=c,if(r==c+1,-c,1))), L=sum(m=1,n+4,(M^0-M)^m/m));L[n+1,k+1]}
Original entry on oeis.org
1, 4, 19, 109, 739, 5779, 51139, 504739, 5494339, 65369539, 843747139, 11741033539, 175200329539, 2790549065539, 47251477577539, 847548190793539, 16053185741897539, 320165936763977539, 6706533708227657539, 147206624680428617539, 3378708717041050697539
Offset: 0
G.f.: A(x) = 1 + 4*x + 19*x^2 + 109*x^3 + 739*x^4 + 5779*x^5 + 51139*x^6 + 504739*x^7 + 5494339*x^8 + 65369539*x^9 + 843747139*x^10 + ...
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[(&+[Factorial(j): j in [2..n+3]])/8: n in [0..30]]; // G. C. Greubel, Sep 05 2022
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a:=n->sum(j!/8,j=2..n): seq(a(n), n=3..21); # Zerinvary Lajos, Jan 08 2007
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Table[Sum[i!/8, {i, 2, n}], {n, 3, 21}] (* Zerinvary Lajos, Jul 12 2009 *)
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{a(n)=1+sum(k=4,n+3,k!)*3/4!}
for(n=0,25,print1(a(n),", "))
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[sum(factorial(j) for j in (2..n+3))/8 for n in (0..30)] # G. C. Greubel, Sep 05 2022
A117399
Column 1 (divided by 2) of triangle A117398, which is the matrix log of A117396.
Original entry on oeis.org
1, 1, 2, 7, 34, 202, 1402, 11122, 99322, 986362, 10784122, 128720122, 1665516922, 23220588922, 347025670522, 5534133996922, 93802153836922, 1683934185324922, 31917365573484922, 636941680764780922, 13348854673487724922
Offset: 1
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m=42;
M= Table[If[k>n-1, 0, If[k==n-1, n, -1]], {n,0,m+1}, {k,0,m+1}];
T:= T= Sum[MatrixPower[M, j]/j, {j,m+1}];
Table[T[[n+1, 2]]/2, {n,2,30}] (* G. C. Greubel, Sep 06 2022 *)
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{a(n)=local(M=matrix(n+4,n+4,r,c,if(r>=c,if(r==c+1,-c,1))), L=sum(m=1,n+4,(M^0-M)^m/m));L[n+2,2]/2}
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