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.

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A318123 a(0) = a(1) = 1; a(n) = Sum_{k=0..n-2} lcm(a(k), a(n-k-2)).

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 11, 22, 42, 90, 165, 364, 710, 1586, 2885, 6288, 12546, 28686, 55406, 130284, 261016, 555716, 1149559, 2552096, 5429610, 11510594, 23650462, 54063580, 109053128, 262968892, 501771275, 1192348336, 2635608642, 5766609158, 11935917602, 28867050172, 58702673480
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 18 2018

Keywords

Links

Crossrefs

Cf. A007463, A007477, A318122.

Programs

  • Mathematica
    a[0] = a[1] = 1; a[n_] := a[n] = Sum[LCM[a[k], a[n - k - 2]], {k, 0, n - 2}]; Table[a[n], {n, 0, 36}]

A354737 a(0) = a(1) = 1; a(n) = n * Sum_{k=0..n-2} a(k) * a(n-k-2).

Original entry on oeis.org

1, 1, 2, 6, 20, 80, 336, 1568, 7584, 39312, 210080, 1180256, 6813312, 40890304, 251528704, 1597332480, 10376040448, 69259146752, 472084038144, 3295588345344, 23459477468160, 170610216311808, 1263629972183040, 9543419750909952, 73322350509367296, 573544008429363200
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 04 2022

Keywords

Crossrefs

Cf. A000699, A007477, A185183, A354738.

Programs

  • Mathematica
    a[0] = a[1] = 1; a[n_] := a[n] = n Sum[a[k] a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 25}]
nmax = 25; A[] = 0; Do[A[x] = 1 + x + 2 x^2 A[x]^2 + 2 x^3 A[x] D[A[x], x] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

Formula

G.f. A(x) satisfies: A(x) = 1 + x + 2 * x^2 * A(x)^2 + 2 * x^3 * A(x) * A'(x).

A354738 a(0) = a(1) = 1; a(n) = (n-1) * Sum_{k=0..n-2} a(k) * a(n-k-2).

Original entry on oeis.org

1, 1, 1, 4, 9, 40, 135, 636, 2688, 13552, 65871, 355520, 1906740, 10963656, 63468171, 386532944, 2383820820, 15294890848, 99626199832, 670333562352, 4583302104450, 32213942456000, 230118463761795, 1683896120829384, 12520330728001670, 95110075114630416
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 04 2022

Keywords

Crossrefs

Cf. A000699, A007477, A185183, A354737.

Programs

  • Mathematica
    a[0] = a[1] = 1; a[n_] := a[n] = (n - 1) Sum[a[k] a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 25}]
nmax = 25; A[] = 0; Do[A[x] = 1 + x + x^2 A[x]^2 + 2 x^3 A[x] D[A[x], x] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

Formula

G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x)^2 + 2 * x^3 * A(x) * A'(x).

A367259 G.f. satisfies A(x) = 1 + x*A(x)^3 * (1 + x*A(x))^2.

Original entry on oeis.org

1, 1, 5, 27, 169, 1138, 8061, 59188, 446455, 3438863, 26935372, 213883631, 1717852129, 13931065117, 113913095218, 938154381748, 7774936633411, 64791892224825, 542598513709481, 4564001359135661, 38541714429405304, 326640923339410701
Offset: 0

Views

Author

Seiichi Manyama, Nov 11 2023

Keywords

Crossrefs

Cf. A007477, A161634, A214372.
Cf. A367232.

Programs

  • Maple
    A367259 := proc(n)
    add(binomial(3*k+(n-k)+1,k) * binomial(2*k,n-k) / (3*k+(n-k)+1),k=0..n) ;
end proc:
seq(A367259(n),n=0..70) ; # R. J. Mathar, Dec 04 2023
  • PARI
    a(n, s=2, t=3, u=1) = sum(k=0, n, binomial(t*k+u*(n-k)+1, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+1));

Formula

If g.f. satisfies A(x) = 1 + x*A(x)^t * (1 + x*A(x)^u)^s, then a(n) = Sum_{k=0..n} binomial(t*k+u*(n-k)+1,k) * binomial(s*k,n-k) / (t*k+u*(n-k)+1).
D-finite with recurrence 8*n*(26345215448853860010445423574*n -20961990363613887876514780359) *(2*n+1)*(2*n-1) *(n+1)*a(n) +8*n*(2*n-1)* (52690430897707720020890847148*n^3 -8177454564962587489822763646077*n^2 +19278991143331529980160099092658*n-11257584930903257675329694643457) *a(n-1) +2*(-268110383402413819740981254825038*n^5 +2815977437639263120434136294300085*n^4 -10349136726006717489413692948200650*n^3 +17659039091779726381787370980047525*n^2 -14385155927699861644653059971375422*n +4528097093401255127907905744957880) *a(n-2) +2*(2433809541139490470204589489378644*n^5 -29695021140710269817720089645612595*n^4 +147295722233051282998786410783007430*n^3 -369735985683645289967183608338045205*n^2 +467786753736867474630837654962591406*n -237792800129696483300250545739991320) *a(n-3) +2*(-2771166843885660051994398777044296*n^5 +70669385063622159693270493531099173*n^4 -615983650096141972053534317661369592*n^3 +2483715780351994976504831765723882733*n^2 -4785455586973998561063713309602358866*n +3584098048545781487176463022333484200) *a(n-4) +(-9719685405660460345432742140418255*n^5 +101488259196839193588678566387929584*n^4 +90729575616085725241944658061815579*n^3 -4613811089886954307541928620224211376*n^2 +19004407111946953332012410754931442164*n -24034005967022354223275806054437127680) *a(n-5) +5*(4577999937824616490204866357596875*n^5 -139866352876176382080407833814299250*n^4 +1615537655758910403049946493111918725*n^3 -8980705919938192198141139371077714070*n^2 +24274376174445463863335689528941329496*n -25684691566228873512769902557616240960) *a(n-6) +30*(1336493817891495200869338759185*n -5552932550849126027962496889033) *(5*n-32)*(5*n-26)*(5*n-29)*(5*n-28)*a(n-7)=0. - R. J. Mathar, Dec 04 2023

A378320 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(2*n-2*r+k,r) * binomial(r,n-r)/(2*n-2*r+k) for k > 0.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 2, 0, 1, 4, 6, 6, 3, 0, 1, 5, 10, 13, 11, 6, 0, 1, 6, 15, 24, 27, 22, 11, 0, 1, 7, 21, 40, 55, 57, 44, 22, 0, 1, 8, 28, 62, 100, 124, 121, 90, 44, 0, 1, 9, 36, 91, 168, 241, 278, 258, 187, 90, 0, 1, 10, 45, 128, 266, 432, 570, 620, 555, 392, 187, 0
Offset: 0

Views

Author

Seiichi Manyama, Nov 23 2024

Keywords

Examples

			Square array begins:
  1,  1,  1,   1,   1,   1,    1, ...
  0,  1,  2,   3,   4,   5,    6, ...
  0,  1,  3,   6,  10,  15,   21, ...
  0,  2,  6,  13,  24,  40,   62, ...
  0,  3, 11,  27,  55, 100,  168, ...
  0,  6, 22,  57, 124, 241,  432, ...
  0, 11, 44, 121, 278, 570, 1077, ...

Crossrefs

Columns k=0..1 give A000007, A007477.
Cf. A071919, A378321.

Programs

  • PARI
    T(n, k, t=0, u=2) = if(k==0, 0^n, k*sum(r=0, n, binomial(t*r+u*(n-r)+k, r)*binomial(r, n-r)/(t*r+u*(n-r)+k)));
matrix(7, 7, n, k, T(n-1, k-1))

Formula

G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x + x^2 * A_k(x)^(2/k) )^k for k > 0.
G.f. of column k: B(x)^k where B(x) is the g.f. of A007477.
B(x)^k = B(x)^(k-1) + x * B(x)^(k-1) + x^2 * B(x)^(k+1). So T(n,k) = T(n,k-1) + T(n-1,k-1) + T(n-2,k+1) for n > 1.

A105847 Inverse binomial transform of number triangle A105632.

Original entry on oeis.org

1, 0, 1, 1, -1, 1, 0, 3, -2, 1, 2, -3, 6, -3, 1, 0, 9, -10, 10, -4, 1, 5, -9, 26, -22, 15, -5, 1, 0, 29, -42, 59, -40, 21, -6, 1, 14, -29, 108, -121, 115, -65, 28, -7, 1, 0, 99, -174, 308, -276, 202, -98, 36, -8, 1, 42, -99, 450, -620, 734, -546, 329, -140, 45, -9, 1, 0, 351, -726, 1547, -1700, 1540, -980, 506, -192, 55, -10, 1, 132
Offset: 0

Views

Author

Paul Barry, Apr 22 2005

Keywords

Comments

Product of Riordan array (1/(1+x),x/(1+x)) (inverse binomial matrix) and A105632. Rows sums are A007477.

Examples

			Triangle begins
1;
0,1;
1,-1,1;
0,3,-2,1;
2,-3,6,-3,1;
0,9,-10,10,-4,1;

Formula

Column k has g.f. (sum{j=0..k-1, C(k-1, j)C(j)x^(2j)/(1-4x^2)^(j+1/2)}+0^k*(1-sqrt(1-4x^2))/(2x^2))(x/(1+x))^k

A247171 G.f.: (2*x^2+4*x+3)/((2*x+2)*sqrt(-4*x^3-4*x^2+1))-1/(2*x+2).

Original entry on oeis.org

1, 1, 3, 4, 11, 21, 48, 106, 235, 535, 1203, 2751, 6272, 14392, 33078, 76224, 176043, 407253, 943833, 2190397, 5090371, 11843689, 27586793, 64320191, 150102784, 350586496, 819477792, 1916861350, 4486760870, 10508582130, 24626700888
Offset: 0

Views

Author

Vladimir Kruchinin, Nov 22 2014

Keywords

Crossrefs

Cf. A007477.

Programs

  • Mathematica
    CoefficientList[Series[(2 x^2 + 4 x + 3) / ((2 x + 2) Sqrt[-4 x^3 - 4 x^2 + 1]) - 1 / (2 x + 2), {x, 0, 40}], x] (* Vincenzo Librandi, Nov 22 2014 *)
  • Maxima
    a(n):=if n=0 then 1 else n*sum((binomial(2*k,n-k)*binomial(n-k-1,k-1))/k,k,1,n);

Formula

a(n) = n*Sum_{k=1..n} (binomial(2*k,n-k)*binomial(n-k-1,k-1))/k, n>0, a(0)=1.
D-finite with recurrence: 3*n*a(n) +(7*n-8)*a(n-1) +2*(-3*n-2)*a(n-2) +2*(-19*n+35)*a(n-3) +2*(-26*n+81)*a(n-4) +4*(-8*n+35)*a(n-5) +4*(-2*n+11)*a(n-6)=0. - R. J. Mathar, Jan 25 2020
Previous Showing 21-27 of 27 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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