A127083
Column 0 and row sums of triangle A127082.
Original entry on oeis.org
1, 1, 2, 5, 16, 64, 308, 1728, 11046, 79065, 625049, 5397939, 50476959, 507435548, 5451145709, 62260278817, 752770290544, 9598571168318, 128651201239737, 1807273852520354, 26541004709809462, 406530038758976731
Offset: 0
-
T[n_, k_]:= T[n, k]= If[k==n, 1, Coefficient[(1 + x*Sum[x^(r-k)*Sum[T[r, c], {c,k,r}], {r,k,n-1}] + x^(n+1))^(k+1), x, n-k]]; Table[T[n, 0], {n,0,25}] (* G. C. Greubel, Jan 30 2020 *)
Original entry on oeis.org
1, 2, 7, 28, 127, 650, 3737, 23996, 170866, 1338578, 11446714, 106063630, 1057817614, 11288886056, 128243813228, 1543828592478, 19616461337281, 262178561430244, 3674568043513202, 53861542554953612, 823710227331537712
Offset: 1
-
T[n_, k_]:= T[n, k]= If[k==n, 1, Coefficient[(1 + x*Sum[x^(r-k)*Sum[T[r, c], {c,k,r}], {r,k,n-1}] + x^(n+1))^(k+1), x, n-k]]; Table[T[n, 1], {n,1,25}] (* G. C. Greubel, Jan 30 2020 *)
Original entry on oeis.org
1, 3, 15, 85, 531, 3600, 26266, 205353, 1716582, 15321056, 145819266, 1477589301, 15908557455, 181553715486, 2190398368254, 27859946518796, 372542199781464, 5223365137285467, 76597458027515272, 1172078722366916586
Offset: 2
-
T[n_, k_]:= T[n, k]= If[k==n, 1, Coefficient[(1 + x*Sum[x^(r-k)*Sum[T[r, c], {c,k,r}], {r,k,n-1}] + x^(n+1))^(k+1), x, n-k]]; Table[T[n, 2], {n,2,25}] (* G. C. Greubel, Jan 30 2020 *)
Original entry on oeis.org
1, 4, 26, 192, 1551, 13416, 122770, 1180496, 11883079, 124992672, 1372811900, 15741602608, 188470662702, 2356327731016, 30760057620142, 419124712458444, 5956905826561685, 88230307480324360, 1360309585677285312
Offset: 3
-
T[n_, k_]:= T[n, k]= If[k==n, 1, Coefficient[(1 + x*Sum[x^(r-k)*Sum[T[r, c], {c,k,r}], {r,k,n-1}] + x^(n+1))^(k+1), x, n-k]]; Table[T[n, 3], {n,3,25}] (* G. C. Greubel, Jan 30 2020 *)
A127090
Central terms of triangle A127082; a(n) = A127082(2*n,n).
Original entry on oeis.org
1, 2, 15, 192, 3635, 92730, 2998366, 117857600, 5465922021, 292505725990, 17755023166100, 1205937035790936, 90649549598544937, 7473077539914412930, 670529221966656416145, 65059053545271098896848
Offset: 0
-
T[n_, k_]:= T[n, k]= If[k==n, 1, Coefficient[(1 + x*Sum[x^(r-k)*Sum[T[r, c], {c,k,r}], {r,k,n-1}] + x^(n+1))^(k+1), x, n-k]]; Table[T[2*n, n], {n,0,15}] (* G. C. Greubel, Jan 30 2020 *)
A127087
Convolution square-root of column 1 (A127084) of triangle A127082.
Original entry on oeis.org
1, 1, 3, 11, 48, 244, 1420, 9318, 68019, 545984, 4772890, 45079020, 456958589, 4943710161, 56809133108, 690510011727, 8845800877774, 119052630071419, 1678622651280617, 24733730857289108, 379989034049167269
Offset: 0
-
T[n_, k_]:= T[n, k]= If[k==n, 1, Coefficient[(1 + x*Sum[x^(r-k)*Sum[T[r, c], {c, k, r}], {r, k, n-1}] + x^(n+1))^(k+1), x, n-k]]; CoefficientList[Series[ (Sum[T[n, 1]*x^n, {n,0,25}]/x)^(1/2), {x,0,20}], x] (* G. C. Greubel, Jan 30 2020 *)
A127088
Convolution cube-root of column 2 (A127085) of triangle A127082.
Original entry on oeis.org
1, 1, 4, 20, 117, 770, 5581, 44023, 375118, 3434312, 33632306, 350894959, 3885892547, 45520247052, 562266198499, 7301972285296, 99436168734138, 1416444089850373, 21059162813775906, 326127491494213657
Offset: 0
-
T[n_, k_]:= T[n, k]= If[k==n, 1, Coefficient[(1 + x*Sum[x^(r-k)*Sum[T[r, c], {c, k, r}], {r, k, n-1}] + x^(n+1))^(k+1), x, n-k]]; CoefficientList[Series[ (Sum[T[n, 2]*x^n, {n,0,25}]/x^2)^(1/3), {x,0,20}], x] (* G. C. Greubel, Jan 30 2020 *)
A127091
Derived from central terms (A127090) of triangle A127082; a(n) = A127090(n)/(n+1).
Original entry on oeis.org
1, 1, 5, 48, 727, 15455, 428338, 14732200, 607324669, 29250572599, 1614093015100, 100494752982578, 6973042276811149, 533791252851029495, 44701948131110427743, 4066190846579443681053, 399302066160095572863595
Offset: 0
-
T[n_, k_]:= T[n, k]= If[k==n, 1, Coefficient[(1 + x*Sum[x^(r-k)*Sum[T[r, c], {c,k,r}], {r,k,n-1}] + x^(n+1))^(k+1), x, n-k]]; Table[T[2*n, n]/(n+1), {n, 0, 15}] (* G. C. Greubel, Jan 30 2020 *)
Showing 1-8 of 8 results.
Comments