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 1-10 of 11 results. Next

A158836 Column 0 of triangle A158835.

Original entry on oeis.org

1, 1, 4, 27, 254, 3062, 45052, 783151, 15712342, 357459042, 9094926988, 255939571048, 7893741230500, 264806871279676, 9600056691219936, 374033821840909263, 15586672520501193866, 691789220336675178652
Offset: 0

Views

Author

Paul D. Hanna, Mar 28 2009

Keywords

Comments

Triangle A158835 transforms diagonals in the array A158825 of coefficients of successive iterations of x*C(x) where C(x) is the Catalan function (A000108).

Links

Crossrefs

Cf. A158835, A158837, A158838, A158839, A158840.

Programs

  • PARI
    {a(n)=local(F=x, CAT=serreverse(x-x^2+x*O(x^(n+2))),M,N,P);M=matrix(n+2, n+2, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, CAT)); polcoeff(F, c)); N=matrix(n+1, n+1, r, c, M[r, c]); P=matrix(n+1, n+1, r, c, M[r+1, c]); (P~*N~^-1)[n+1,1]}

A158837 Column 1 of triangle A158835.

Original entry on oeis.org

1, 2, 11, 94, 1072, 15212, 257777, 5074738, 113775490, 2861365660, 79763482974, 2440866020252, 81343355108428, 2932370770780016, 113695507437209845, 4717853729131352186, 208615291319607614600, 9792578421235713418464
Offset: 0

Views

Author

Paul D. Hanna, Mar 28 2009

Keywords

Comments

Triangle A158835 transforms diagonals in the array A158825 of coefficients of successive iterations of x*C(x) where C(x) is the Catalan function (A000108).

Crossrefs

Cf. A158835, A158836, A158838, A158839, A158840.

Programs

  • PARI
    {a(n)=local(F=x, CAT=serreverse(x-x^2+x*O(x^(n+3))),M,N,P);M=matrix(n+3, n+3, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, CAT)); polcoeff(F, c)); N=matrix(n+2, n+2, r, c, M[r, c]); P=matrix(n+2, n+2, r, c, M[r+1, c]); (P~*N~^-1)[n+2,2]}

A158839 Column 3 of triangle A158835.

Original entry on oeis.org

1, 4, 34, 412, 6325, 116372, 2483706, 60168736, 1628677692, 48672911296, 1590752204044, 56418074957840, 2157411204773415, 88464995576660084, 3871611011946560294, 180101399407072883012, 8873328068327122625596
Offset: 0

Views

Author

Paul D. Hanna, Mar 28 2009

Keywords

Comments

Triangle A158835 transforms diagonals in the array A158825 of coefficients of successive iterations of x*C(x) where C(x) is the Catalan function (A000108).

Crossrefs

Cf. A158835, A158836, A158837, A158836, A158840.

Programs

  • PARI
    {a(n)=local(F=x, CAT=serreverse(x-x^2+x*O(x^(n+5))),M,N,P);M=matrix(n+5, n+5, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, CAT)); polcoeff(F, c)); N=matrix(n+4, n+4, r, c, M[r, c]); P=matrix(n+4, n+4, r, c, M[r+1, c]); (P~*N~^-1)[n+4,4]}

A158840 Row sums of triangle A158835.

Original entry on oeis.org

1, 2, 7, 42, 374, 4391, 63637, 1095362, 21823226, 493898216, 12515588806, 351062669154, 10798972965266, 361471373319171, 13080119556342713, 508813238759275712, 21174032937728251318, 938646693399848483498
Offset: 0

Views

Author

Paul D. Hanna, Mar 28 2009

Keywords

Comments

Triangle A158835 transforms diagonals in the array A158825 of coefficients of successive iterations of x*C(x) where C(x) is the Catalan function (A000108).

Crossrefs

Cf. A158835, A158836, A158837, A158838, A158839.

Programs

  • PARI
    {a(n)=local(F=x, CAT=serreverse(x-x^2+x*O(x^(n+2))), M, N, P); M=matrix(n+2, n+2, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, CAT)); polcoeff(F, c)); N=matrix(n+1, n+1, r, c, M[r, c]); P=matrix(n+1, n+1, r, c, M[r+1, c]); sum(k=0,n,(P~*N~^-1)[n+1, k+1])}

A158838 Column 2 of triangle A158835.

Original entry on oeis.org

1, 3, 21, 217, 2904, 47337, 906557, 19910808, 492818850, 13564326950, 410807572044, 13573135469214, 485765085176420, 18717987193565613, 772565258231236269, 34002334709760133807, 1589555183231724515700
Offset: 0

Views

Author

Paul D. Hanna, Mar 28 2009

Keywords

Comments

Triangle A158835 transforms diagonals in the array A158825 of coefficients of successive iterations of x*C(x) where C(x) is the Catalan function (A000108).

Crossrefs

Cf. A158835, A158836, A158837, A158839, A158840.

Programs

  • PARI
    {a(n)=local(F=x, CAT=serreverse(x-x^2+x*O(x^(n+4))),M,N,P);M=matrix(n+4, n+4, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, CAT)); polcoeff(F, c)); N=matrix(n+3, n+3, r, c, M[r, c]); P=matrix(n+3, n+3, r, c, M[r+1, c]); (P~*N~^-1)[n+3,3]}

A158825 Square array of coefficients in the successive iterations of x*C(x) = (1-sqrt(1-4*x))/2 where C(x) is the g.f. of the Catalan numbers (A000108); read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 6, 5, 1, 4, 12, 21, 14, 1, 5, 20, 54, 80, 42, 1, 6, 30, 110, 260, 322, 132, 1, 7, 42, 195, 640, 1310, 1348, 429, 1, 8, 56, 315, 1330, 3870, 6824, 5814, 1430, 1, 9, 72, 476, 2464, 9380, 24084, 36478, 25674, 4862, 1, 10, 90, 684, 4200, 19852, 67844, 153306, 199094, 115566, 16796
Offset: 1

Views

Author

Paul D. Hanna, Mar 28 2009, Mar 29 2009

Keywords

Examples

			Square array of coefficients in iterations of x*C(x) begins:
  1,  1,   2,    5,    14,      42,      132,       429,       1430, ... A000108;
  1,  2,   6,   21,    80,     322,     1348,      5814,      25674, ... A121988;
  1,  3,  12,   54,   260,    1310,     6824,     36478,     199094, ... A158826;
  1,  4,  20,  110,   640,    3870,    24084,    153306,     993978, ... A158827;
  1,  5,  30,  195,  1330,    9380,    67844,    500619,    3755156, ... A158828;
  1,  6,  42,  315,  2464,   19852,   163576,   1372196,   11682348, ...;
  1,  7,  56,  476,  4200,   38052,   351792,   3305484,   31478628, ...;
  1,  8,  72,  684,  6720,   67620,   693048,   7209036,   75915708, ...;
  1,  9,  90,  945, 10230,  113190,  1273668,  14528217,  167607066, ...;
  1, 10, 110, 1265, 14960,  180510,  2212188,  27454218,  344320262, ...;
  1, 11, 132, 1650, 21164,  276562,  3666520,  49181418,  666200106, ...;
  1, 12, 156, 2106, 29120,  409682,  5841836,  84218134, 1225314662, ...;
  1, 13, 182, 2639, 39130,  589680,  8999172, 138755799, 2157976392, ...;
  1, 14, 210, 3255, 51520,  827960, 13464752, 221101608, 3660331064, ...;
  1, 15, 240, 3960, 66640, 1137640, 19640032, 342179672, 6007747368, ...;
  1, 16, 272, 4760, 84864, 1533672, 28012464, 516105720, 9578580504, ...;
ILLUSTRATE ITERATIONS.
       Let G(x) = x*C(x), then the first few iterations of G(x) are:
           G(x) = x +   x^2 +  2*x^3 +   5*x^4 +  14*x^5 + ...;
        G(G(x)) = x + 2*x^2 +  6*x^3 +  21*x^4 +  80*x^5 + ...;
     G(G(G(x))) = x + 3*x^2 + 12*x^3 +  54*x^4 + 260*x^5 + ...;
  G(G(G(G(x)))) = x + 4*x^2 + 20*x^3 + 110*x^4 + 640*x^5 + ...;
...
RELATED TRIANGLES.
The g.f. of column n is (g.f. of row n of A158830)/(1-x)^n
where triangle A158830 begins: 1;
      1,       0;
      2,       0,        0;
      5,       1,        0,        0;
     14,      10,        0,        0,       0;
     42,      70,        8,        0,       0,       0;
    132,     424,      160,        4,       0,       0,     0;
    429,    2382,     1978,      250,       1,       0,     0,   0;
   1430,   12804,    19508,     6276,     302,       0,     0,   0, 0;
   4862,   66946,   168608,   106492,   15674,     298,     0,   0, 0, 0;
  16796,  343772,  1337684,  1445208,  451948,   33148,   244,   0, 0, 0, 0;
  58786, 1744314, 10003422, 16974314, 9459090, 1614906, 61806, 162, 0, 0, 0, 0;
  ...
Triangle A158835 transforms one diagonal into the next:
       1;
       1,      1;
       4,      2,     1;
      27,     11,     3,    1;
     254,     94,    21,    4,   1;
    3062,   1072,   217,   34,   5,  1;
   45052,  15212,  2904,  412,  50,  6, 1;
  783151, 257777, 47337, 6325, 695, 69, 7, 1; ...
so that:
  A158835 * A158831 = A158832;
  A158835 * A158832 = A158833;
  A158835 * A158833 = A158834;
where the diagonals start:
  A158831 = [1, 1,  6,  54,  640,  9380,  163576,  3305484, ...];
  A158832 = [1, 2, 12, 110, 1330, 19852,  351792,  7209036, ...];
  A158833 = [1, 3, 20, 195, 2464, 38052,  693048, 14528217, ...];
  A158834 = [1, 4, 30, 315, 4200, 67620, 1273668, 27454218, ...].

Links

Crossrefs

Rows: A000108, A121988, A158826, A158827, A158828.
Columns: A000012, A000027, A002378, A160378.
Antidiagonal sums: A158829.
Diagonals: A158831, A158832, A158833, A158834.
Related triangles: A158830, A158835.
Variant: A122888.

Programs

  • Mathematica
    Clear[row]; nmax = 12;
row[n_]:= row[n]= CoefficientList[Nest[(1-Sqrt[1-4#])/2&, x, n] + O[x]^(nmax+1), x] //Rest;
T[n_, k_]:= row[n][[k]];
Table[T[n-k+1, k], {n, nmax}, {k, n}]//Flatten (* Jean-François Alcover, Jul 13 2018, updated Aug 09 2018 *)
  • PARI
    {T(n,k)= local(F=serreverse(x-x^2+O(x^(k+2))), G=x);
for(i=1, n, G=subst(F,x,G)); polcoeff(G,k)}

Formula

G.f. of column n = (g.f. of row n of A158830)/(1-x)^n.
Row k equals the first column of the k-th matrix power of Catalan triangle A033184; thus triangle A033184 transforms row n into row n+1 of this array (A158825). - Paul D. Hanna, Mar 30 2009
From G. C. Greubel, Apr 01 2021: (Start)
T(n, 1) = A000012(n), T(n, 2) = A000027(n).
T(n, 3) = A002378(n), T(n, 4) = A160378(n+1). (End)

A158831 A diagonal in the array A158825 of coefficients of successive iterations of x*C(x), where C(x) is the Catalan function (A000108).

Original entry on oeis.org

1, 1, 6, 54, 640, 9380, 163576, 3305484, 75915708, 1952409954, 55573310936, 1734182983962, 58863621238500, 2159006675844616, 85088103159523296, 3585740237981536700, 160894462797493581048, 7658326127259130753070
Offset: 1

Views

Author

Paul D. Hanna, Mar 28 2009

Keywords

Comments

Triangle A158835 transforms this sequence into A158832, the next diagonal in A158825.

Examples

			Table of coefficients in the i-th iteration of x*Catalan(x):
(1);
1,(1),2,5,14,42,132,429,1430,4862,16796,58786,208012,...;
1,2,(6),21,80,322,1348,5814,25674,115566,528528,2449746,...;
1,3,12,(54),260,1310,6824,36478,199094,1105478,6227712,...;
1,4,20,110,(640),3870,24084,153306,993978,6544242,43652340,...;
1,5,30,195,1330,(9380),67844,500619,3755156,28558484,...;
1,6,42,315,2464,19852,(163576),1372196,11682348,100707972,...;
1,7,56,476,4200,38052,351792,(3305484),31478628,303208212,...;
1,8,72,684,6720,67620,693048,7209036,(75915708),807845676,...;
1,9,90,945,10230,113190,1273668,14528217,167607066,(1952409954),...; ...
where terms in parenthesis form the initial terms of this sequence.

Crossrefs

Cf. A158825, A158832, A158833, A158834.

Programs

  • Mathematica
    nmax = 18;
g[x_] := Module[{y}, Expand[Normal[(1 - Sqrt[1 - 4*y])/2 + O[y]^(nmax+2)] /. y -> x][[1 ;; nmax+1]]];
T = Table[Nest[g, x, n] // CoefficientList[#, x]& // Rest, {n, 1, nmax+1}];
Prepend[Diagonal[T, 1], 1] (* Jean-François Alcover, Jul 13 2018 *)
  • PARI
    {a(n)=local(F=serreverse(x-x^2+O(x^(n+2))),G=x); for(i=1,n-1,G=subst(F,x,G));polcoeff(G,n)}

A158832 Main diagonal in the array A158825 of coefficients of successive iterations of x*C(x), where C(x) is the Catalan function (A000108).

Original entry on oeis.org

1, 2, 12, 110, 1330, 19852, 351792, 7209036, 167607066, 4357308098, 125219900520, 3941126688798, 134808743674176, 4979127855477336, 197480359402576304, 8370550907396970684, 377599345119560766534, 18061714498169627460982
Offset: 1

Views

Author

Paul D. Hanna, Mar 28 2009

Keywords

Comments

Triangle A158835 transforms A158831 into this sequence, where A158831 is the previous diagonal in A158825.
Triangle A158835 transforms this sequence into A158833, the next diagonal in A158825.

Examples

			Array of coefficients in the i-th iteration of x*Catalan(x):
(1),1,2,5,14,42,132,429,1430,4862,16796,58786,208012,...;
1,(2),6,21,80,322,1348,5814,25674,115566,528528,2449746,...;
1,3,(12),54,260,1310,6824,36478,199094,1105478,6227712,...;
1,4,20,(110),640,3870,24084,153306,993978,6544242,43652340,...;
1,5,30,195,(1330),9380,67844,500619,3755156,28558484,...;
1,6,42,315,2464,(19852),163576,1372196,11682348,100707972,...;
1,7,56,476,4200,38052,(351792),3305484,31478628,303208212,...;
1,8,72,684,6720,67620,693048,(7209036),75915708,807845676,...;
1,9,90,945,10230,113190,1273668,14528217,(167607066),...;
1,10,110,1265,14960,180510,2212188,27454218,344320262,(4357308098),...; ...
where terms in parenthesis form the initial terms of this sequence.

Links

Crossrefs

Cf. A158825, A158831, A158833, A158834.

Programs

  • Mathematica
    a[n_] := Module[{x, F, G}, F = InverseSeries[x - x^2 + O[x]^(n+2)]; G = x; For[i = 1, i <= n, i++, G = (F /. x -> G)]; Coefficient[G, x, n]];
Array[a, 18] (* Jean-François Alcover, Jul 13 2018, from PARI *)
  • PARI
    {a(n)=local(F=serreverse(x-x^2+O(x^(n+2))),G=x); for(i=1,n,G=subst(F,x,G));polcoeff(G,n)}

A158833 A diagonal in the array A158825 of coefficients of successive iterations of x*C(x), where C(x) is the Catalan function (A000108).

Original entry on oeis.org

1, 3, 20, 195, 2464, 38052, 693048, 14528217, 344320262, 9100230282, 265305808404, 8456446272144, 292528760419440, 10913859037065560, 436812586581170976, 18668379209883807385, 848499254768957476312
Offset: 1

Views

Author

Paul D. Hanna, Mar 28 2009

Keywords

Comments

Triangle A158835 transforms A158832 into this sequence, where A158832 is the previous diagonal in A158825.
Triangle A158835 transforms this sequence into A158834, the next diagonal in A158825.

Examples

			Array of coefficients in the i-th iteration of x*Catalan(x):
1,1,2,5,14,42,132,429,1430,4862,16796,58786,208012,...;
(1),2,6,21,80,322,1348,5814,25674,115566,528528,2449746,...;
1,(3),12,54,260,1310,6824,36478,199094,1105478,6227712,...;
1,4,(20),110,640,3870,24084,153306,993978,6544242,43652340,...;
1,5,30,(195),1330,9380,67844,500619,3755156,28558484,...;
1,6,42,315,(2464),19852,163576,1372196,11682348,100707972,...;
1,7,56,476,4200,(38052),351792,3305484,31478628,303208212,...;
1,8,72,684,6720,67620,(693048),7209036,75915708,807845676,...;
1,9,90,945,10230,113190,1273668,(14528217),167607066,...;
1,10,110,1265,14960,180510,2212188,27454218,(344320262),...;
1,11,132,1650,21164,276562,3666520,49181418,666200106,(9100230282),...; ...
where terms in parenthesis form the initial terms of this sequence.

Crossrefs

Cf. A158825, A158831, A158832, A158834.

Programs

  • Mathematica
    a[n_] := Module[{x, F, G}, F = InverseSeries[x - x^2 + O[x]^(n+2)]; G = x; For[i = 1, i <= n+1, i++, G = (F /. x -> G)]; Coefficient[G, x, n]];
Array[a, 17] (* Jean-François Alcover, Jul 13 2018, from PARI *)
  • PARI
    {a(n)=local(F=serreverse(x-x^2+O(x^(n+2))),G=x); for(i=1,n+1,G=subst(F,x,G));polcoeff(G,n)}

A158834 A diagonal in the array A158825 of coefficients of successive iterations of x*C(x), where C(x) is the Catalan function (A000108).

Original entry on oeis.org

1, 4, 30, 315, 4200, 67620, 1273668, 27454218, 666200106, 17968302638, 533188477536, 17261808531552, 605452449574320, 22870569475477112, 925663441858807096, 39964465820186753753, 1833332492818402014474
Offset: 1

Views

Author

Paul D. Hanna, Mar 28 2009

Keywords

Comments

Triangle A158835 transforms A158833 into this sequence, where A158833 is the previous diagonal in A158825.

Examples

			Array of coefficients in the i-th iteration of x*Catalan(x):
1,1,2,5,14,42,132,429,1430,4862,16796,58786,208012,...;
1,2,6,21,80,322,1348,5814,25674,115566,528528,2449746,...;
(1),3,12,54,260,1310,6824,36478,199094,1105478,6227712,...;
1,(4),20,110,640,3870,24084,153306,993978,6544242,43652340,...;
1,5,(30),195,1330,9380,67844,500619,3755156,28558484,...;
1,6,42,(315),2464,19852,163576,1372196,11682348,100707972,...;
1,7,56,476,(4200),38052,351792,3305484,31478628,303208212,...;
1,8,72,684,6720,(67620),693048,7209036,75915708,807845676,...;
1,9,90,945,10230,113190,(1273668),14528217,167607066,...;
1,10,110,1265,14960,180510,2212188,(27454218),344320262,...;
1,11,132,1650,21164,276562,3666520,49181418,(666200106),...;
1,12,156,2106,29120,409682,5841836,84218134,1225314662,(17968302638),...; ...
where terms in parenthesis form the initial terms of this sequence.

Crossrefs

Cf. A158825, A158831, A158832, A158833.

Programs

  • Mathematica
    a[n_] := Module[{x, F, G}, F = InverseSeries[x - x^2 + O[x]^(n+2)]; G = x; For[i = 1, i <= n+2, i++, G = (F /. x -> G)]; Coefficient[G, x, n]];
Array[a, 17] (* Jean-François Alcover, Jul 13 2018, from PARI *)
  • PARI
    {a(n)=local(F=serreverse(x-x^2+O(x^(n+2))),G=x); for(i=1,n+2,G=subst(F,x,G));polcoeff(G,n)}
Showing 1-10 of 11 results. Next

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