cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

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

Views

Author

Paul D. Hanna, Jan 05 2007

Keywords

Crossrefs

Programs

  • 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 *)

A127128 Column 1 of triangle A127126.

Original entry on oeis.org

1, 2, 9, 54, 412, 3834, 42131, 533558, 7645065, 122177706, 2153221318, 41464853266, 865908079369, 19484990264956, 469910189792853, 12089047867952058, 330423404118495975, 9561012695542004496
Offset: 1

Views

Author

Paul D. Hanna, Jan 05 2007

Keywords

Comments

Convolution square of A127131.

Crossrefs

Cf. A127126; A127131; other columns: A127127, A127129, A127130.

Programs

  • 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 *)

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

Views

Author

Paul D. Hanna, Jan 05 2007

Keywords

Comments

This is a variant of triangles: A127082, A124328.

Examples

			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; ...
		

Crossrefs

Central terms: A127134.
Variants: A127082, A124328.

Programs

  • 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 *)
  • 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))}

A127127 Column 0 of triangle A127126.

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

Views

Author

Paul D. Hanna, Jan 05 2007

Keywords

Comments

For n > 0, equals one-half of the row sums of triangle A127126.

Crossrefs

Programs

  • 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 *)

A127130 Column 3 of triangle A127126.

Original entry on oeis.org

1, 4, 30, 284, 3217, 42100, 621936, 10206956, 183946656, 3608229588, 76499111202, 1742909621024, 42465280193704, 1101811478508828, 30331386700718440, 883010853509597608, 27105284242369828508, 874996700615422755068
Offset: 3

Views

Author

Paul D. Hanna, Jan 05 2007

Keywords

Comments

Self-convolution 4th power of A127133.

Crossrefs

Cf. A127126; A127133; other columns: A127127, A127128, A127129.

Programs

  • 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 *)

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

Views

Author

Paul D. Hanna, Jan 05 2007

Keywords

Crossrefs

Programs

  • 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 *)

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

Views

Author

Paul D. Hanna, Jan 05 2007

Keywords

Crossrefs

Programs

  • 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 *)

A127134 Central terms of triangle A127126; a(n) = A127126(2n,n).

Original entry on oeis.org

1, 2, 18, 284, 6680, 211398, 8439235, 407247048, 23056215138, 1498169721930, 109876657252604, 8976437481923520, 808257688877060396, 79516093326076500590, 8485004019719253675540, 976009472808194554659440
Offset: 0

Views

Author

Paul D. Hanna, Jan 05 2007

Keywords

Comments

a(n) is divisible by (n+1): a(n)/(n+1) = A127135(n).

Crossrefs

Programs

  • 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 *)

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

Views

Author

Paul D. Hanna, Jan 05 2007

Keywords

Crossrefs

Programs

  • 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 *)

Formula

a(n) = A127134(n)/(n+1).
Showing 1-9 of 9 results.