| Displaying 1-9 of 9 results found.
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1, 1, 5, 36, 336, 3793, 49691, 736301, 12130141, 219488417, 4322334090, 91974793971, 2102457339356, 51377007363853, 1336508757460743, 36876168645675673, 1075680625224925835, 33076997985647151025
MATHEMATICA
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 *)
1, 2, 9, 54, 412, 3834, 42131, 533558, 7645065, 122177706, 2153221318, 41464853266, 865908079369, 19484990264956, 469910189792853, 12089047867952058, 330423404118495975, 9561012695542004496
MATHEMATICA
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, 1], {n, 1, 20}] (* G. C. Greubel, Jan 28 2020 *)
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.
+10
10
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
EXAMPLE
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; ...
MATHEMATICA
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 *)
PROG
(PARI) {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))}
1, 1, 3, 13, 77, 587, 5484, 60582, 771261, 11102828, 178144861, 3149976426, 60825085447, 1273060083700, 28700081677767, 693217471426114, 17857152401368800, 488620956679818191, 14152040894854881662, 432509671322583878614, 13908794132963653028146
COMMENTS
For n > 0, equals one-half of the row sums of triangle A127126.
MATHEMATICA
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 *)
1, 4, 30, 284, 3217, 42100, 621936, 10206956, 183946656, 3608229588, 76499111202, 1742909621024, 42465280193704, 1101811478508828, 30331386700718440, 883010853509597608, 27105284242369828508, 874996700615422755068
COMMENTS
Self-convolution 4th power of A127133.
MATHEMATICA
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, 3], {n, 3, 20}] (* G. C. Greubel, Jan 28 2020 *)
Self-convolution square-root of column 1 ( A127128) of triangle A127126.
+10
7
1, 1, 4, 23, 175, 1650, 18451, 237703, 3457763, 55967155, 996755108, 19360232181, 407152004331, 9215091412811, 223307281633261, 5768104533416742, 158197552561322216, 4591028199312877166, 140551293414196198297
MATHEMATICA
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 *)
1, 1, 6, 52, 576, 7591, 114365, 1923185, 35541761, 714104502, 15475682769, 359547718332, 8911727170149, 234697278951915, 6544781944957233, 192669771715328227, 5971713743277322517, 194402722591654350978
MATHEMATICA
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 *)
1, 2, 18, 284, 6680, 211398, 8439235, 407247048, 23056215138, 1498169721930, 109876657252604, 8976437481923520, 808257688877060396, 79516093326076500590, 8485004019719253675540, 976009472808194554659440
COMMENTS
a(n) is divisible by (n+1): a(n)/(n+1) = A127135(n).
MATHEMATICA
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, 0, 20}] (* G. C. Greubel, Jan 28 2020 *)
1, 1, 6, 71, 1336, 35233, 1205605, 50905881, 2561801682, 149816972193, 9988787022964, 748036456826960, 62173668375158492, 5679720951862607185, 565666934647950245036, 61000592050512159666215
MATHEMATICA
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 *)
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