A127135
Derived from the central terms (A127134) of triangle A127126; a(n) = A127126(2n,n)/(n+1).
Original entry on oeis.org
1, 1, 6, 71, 1336, 35233, 1205605, 50905881, 2561801682, 149816972193, 9988787022964, 748036456826960, 62173668375158492, 5679720951862607185, 565666934647950245036, 61000592050512159666215
Offset: 0
-
T[n_, k_]:= T[n, k]= If[k==n, 1, Coefficient[(1 +x*Sum[x^(r-k-1)*Sum[T[r, c], {c,k+1,r}], {r,k+1,n}] +x^(n+1))^(k+1), x, n-k]]; Table[T[2*n, n]/(n+1), {n, 0, 20}] (* G. C. Greubel, Jan 28 2020 *)
A127126
Triangle, read by rows, where the g.f. of column k, C_k(x), is defined by the recurrence: C_k(x) = [ 1 + Sum_{n>=k+1} C_n(x)*x^(n-k) ]^(k+1) for k>=0.
Original entry on oeis.org
1, 1, 1, 3, 2, 1, 13, 9, 3, 1, 77, 54, 18, 4, 1, 587, 412, 139, 30, 5, 1, 5484, 3834, 1314, 284, 45, 6, 1, 60582, 42131, 14658, 3217, 505, 63, 7, 1, 771261, 533558, 188012, 42100, 6680, 818, 84, 8, 1, 11102828, 7645065, 2721462, 621936, 100621, 12387, 1239, 108, 9, 1
Offset: 0
C_k = [ 1 + x*C_{k+1} + x^2*C_{k+2} + x^3*C_{k+3} +... ]^(k+1).
The columns are generated by working backwards:
C_3 = [ 1 + x*C_4 + x^2*C_5 + x^3*C_6 + x^4*C_7 +... ]^4;
C_2 = [ 1 + x*C_3 + x^2*C_4 + x^3*C_5 + x^4*C_6 +... ]^3;
C_1 = [ 1 + x*C_2 + x^2*C_3 + x^3*C_4 + x^4*C_5 +... ]^2;
C_0 = [ 1 + x*C_1 + x^2*C_2 + x^3*C_3 + x^4*C_4 +... ]^1.
The triangle begins:
1;
1, 1;
3, 2, 1;
13, 9, 3, 1;
77, 54, 18, 4, 1;
587, 412, 139, 30, 5, 1;
5484, 3834, 1314, 284, 45, 6, 1;
60582, 42131, 14658, 3217, 505, 63, 7, 1;
771261, 533558, 188012, 42100, 6680, 818, 84, 8, 1;
11102828, 7645065, 2721462, 621936, 100621, 12387, 1239, 108, 9, 1; ...
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T[n_, k_]:= T[n, k]= If[k==n, 1, Coefficient[(1 + x*Sum[x^(r-k-1)*Sum[T[r, c], {c,k+1,r}], {r,k+1,n}] +x^(n+1))^(k+1), x, n-k]]; Table[T[n, k], {n,0,12}, {k, 0,n}]//Flatten (* G. C. Greubel, Jan 27 2020 *)
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{T(n,k) = if(n==k,1,polcoeff( (1 + x*sum(r=k+1,n,x^(r-k-1)*sum(c=k+1,r, T(r,c))) +x*O(x^n))^(k+1),n-k))}
Original entry on oeis.org
1, 1, 3, 13, 77, 587, 5484, 60582, 771261, 11102828, 178144861, 3149976426, 60825085447, 1273060083700, 28700081677767, 693217471426114, 17857152401368800, 488620956679818191, 14152040894854881662, 432509671322583878614, 13908794132963653028146
Offset: 0
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T[n_, k_]:= T[n, k]= If[k==n, 1, Coefficient[(1 +x*Sum[x^(r-k-1)*Sum[T[r, c], {c,k+1,r}], {r,k+1,n}] + x^(n+1))^(k+1), x, n-k]]; Table[T[n, 0], {n, 0, 20}] (* G. C. Greubel, Jan 28 2020 *)
A127131
Self-convolution square-root of column 1 (A127128) of triangle A127126.
Original entry on oeis.org
1, 1, 4, 23, 175, 1650, 18451, 237703, 3457763, 55967155, 996755108, 19360232181, 407152004331, 9215091412811, 223307281633261, 5768104533416742, 158197552561322216, 4591028199312877166, 140551293414196198297
Offset: 0
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T[n_, k_]:= T[n, k]= If[k==n, 1, Coefficient[(1 +x*Sum[x^(r-k-1)*Sum[T[r, c], {c,k+1,r}], {r,k+1,n}] +x^(n+1))^(k+1), x, n-k]]; CoefficientList[Series[ (Sum[T[n, 1]*x^n, {n,0,22}]/x)^(1/2), {x,0,20}], x] (* G. C. Greubel, Jan 28 2020 *)
A127132
Self-convolution cube-root of column 2 (A127129) of triangle A127126.
Original entry on oeis.org
1, 1, 5, 36, 336, 3793, 49691, 736301, 12130141, 219488417, 4322334090, 91974793971, 2102457339356, 51377007363853, 1336508757460743, 36876168645675673, 1075680625224925835, 33076997985647151025
Offset: 0
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T[n_, k_]:= T[n, k]= If[k==n, 1, Coefficient[(1 +x*Sum[x^(r-k-1)*Sum[T[r, c], {c,k+1,r}], {r,k+1,n}] +x^(n+1))^(k+1), x, n-k]]; CoefficientList[Series[ (Sum[T[n, 2]*x^n, {n,0,22}]/x^2)^(1/3), {x,0,20}], x] (* G. C. Greubel, Jan 28 2020 *)
A127133
Self-convolution 4th root of column 3 (A127130) of triangle A127126.
Original entry on oeis.org
1, 1, 6, 52, 576, 7591, 114365, 1923185, 35541761, 714104502, 15475682769, 359547718332, 8911727170149, 234697278951915, 6544781944957233, 192669771715328227, 5971713743277322517, 194402722591654350978
Offset: 0
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T[n_, k_]:= T[n, k]= If[k==n, 1, Coefficient[(1 +x*Sum[x^(r-k-1)*Sum[T[r, c], {c,k+1,r}], {r,k+1,n}] +x^(n+1))^(k+1), x, n-k]]; CoefficientList[Series[ (Sum[T[n, 3]*x^n, {n,0,25}]/x^3)^(1/4), {x,0,20}], x] (* G. C. Greubel, Jan 28 2020 *)
Showing 1-6 of 6 results.
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