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.

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

Views

Author

Paul D. Hanna, Jan 04 2007

Keywords

Crossrefs

Programs

  • Mathematica
    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 *)

Formula

a(n) = A127082(2*n,n)/(n+1).

A127082 Triangle, read by rows, where the g.f. of column k, C_k(x), is defined by the recursion: C_k(x) = ( 1 + Sum_{n>=1} x^n*C_{n-1+k}(x) )^(k+1).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 5, 7, 3, 1, 16, 28, 15, 4, 1, 64, 127, 85, 26, 5, 1, 308, 650, 531, 192, 40, 6, 1, 1728, 3737, 3600, 1551, 365, 57, 7, 1, 11046, 23996, 26266, 13416, 3635, 620, 77, 8, 1, 79065, 170866, 205353, 122770, 38556, 7356, 973, 100, 9, 1
Offset: 0

Views

Author

Paul D. Hanna, Jan 04 2007

Keywords

Comments

This is a variant of triangle A124328.

Examples

			C_k = [ 1 + x*C_k + x^2*C_{k+1} + x^3*C_{k+2} +... ]^(k+1).
The columns are generated by working backwards:
C_3 = [ 1 + x*C_3 + x^2*C_4 + x^3*C_5 + x^4*C_6 +... ]^4;
C_2 = [ 1 + x*C_2 + x^2*C_3 + x^3*C_4 + x^4*C_5 +... ]^3;
C_1 = [ 1 + x*C_1 + x^2*C_2 + x^3*C_3 + x^4*C_4 +... ]^2;
C_0 = [ 1 + x*C_0 + x^2*C_1 + x^3*C_2 + x^4*C_3 +... ]^1;
thus the row sums equal column 0 shift left.
The triangle begins:
       1;
       1,       1;
       2,       2,       1;
       5,       7,       3,       1;
      16,      28,      15,       4,      1;
      64,     127,      85,      26,      5,     1;
     308,     650,     531,     192,     40,     6,     1;
    1728,    3737,    3600,    1551,    365,    57,     7,    1;
   11046,   23996,   26266,   13416,   3635,   620,    77,    8,   1;
   79065,  170866,  205353,  122770,  38556,  7356,   973,  100,   9,  1;
  625049, 1338578, 1716582, 1180496, 429515, 92730, 13412, 1440, 126, 10, 1;
		

Crossrefs

Cf. variant: A124328;
Columns: A127083, A127084, A127085, A127086, A127090 (central terms).

Programs

  • Mathematica
    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, k], {n,0,12}, {k, 0,n}]//Flatten (* G. C. Greubel, Jan 30 2020 *)
  • PARI
    {T(n,k)=if(n==k,1,polcoeff( (1 + x*sum(r=k,n-1,x^(r-k)*sum(c=k,r, T(r,c) ))+x*O(x^n))^(k+1),n-k))}

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

Views

Author

Paul D. Hanna, Jan 04 2007

Keywords

Crossrefs

Programs

  • Mathematica
    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 *)

A127084 Column 1 of triangle A127082.

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

Views

Author

Paul D. Hanna, Jan 04 2007

Keywords

Comments

Convolution square of A127087.

Crossrefs

Programs

  • Mathematica
    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 *)

A127085 Column 2 of triangle A127082.

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

Views

Author

Paul D. Hanna, Jan 04 2007

Keywords

Comments

Convolution cube of A127088.

Crossrefs

Programs

  • Mathematica
    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 *)

A127086 Column 3 of triangle A127082.

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

Views

Author

Paul D. Hanna, Jan 04 2007

Keywords

Comments

Convolution 4th power of A127089.

Crossrefs

Programs

  • Mathematica
    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 *)

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

Views

Author

Paul D. Hanna, Jan 04 2007

Keywords

Crossrefs

Programs

  • Mathematica
    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

Views

Author

Paul D. Hanna, Jan 04 2007

Keywords

Crossrefs

Programs

  • Mathematica
    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 *)

A127089 Convolution 4th root of column 3 (A127086) of triangle A127082.

Original entry on oeis.org

1, 1, 5, 32, 239, 1981, 17757, 169765, 1717730, 18311250, 205075693, 2408303246, 29611689597, 380712483013, 5111573917042, 71576222215342, 1043901890068909, 15835797676490439, 249530033466698385, 4078781718332965858
Offset: 0

Views

Author

Paul D. Hanna, Jan 04 2007

Keywords

Crossrefs

Programs

  • Mathematica
    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, 3]*x^n, {n,0,25}]/x^3)^(1/4), {x,0,20}], x] (* G. C. Greubel, Jan 30 2020 *)
Showing 1-9 of 9 results.