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.

A319924 a(n) = A143565(2n,n) for n > 0, a(0) = 1.

Original entry on oeis.org

1, 3, 13, 61, 281, 1261, 5545, 24025, 102961, 437581, 1847561, 7759753, 32449873, 135207801, 561632401, 2326762801, 9617286241, 39671305741, 163352435401, 671560012201, 2756930576401, 11303415363241, 46290177201841, 189368906734801, 773942488394401
Offset: 0

Views

Author

Alois P. Heinz, Oct 01 2018

Keywords

Links

Crossrefs

Cf. A002457, A143565.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<2, 2*n+1,
      ((15*n^2-29*n+10)*a(n-1)-(6*n-2)*(2*n-3)*
       a(n-2))/((n-1)*(3*n-4)))
    end:
seq(a(n), n=0..30);

Formula

a(n) = A143565(2n,n) for n > 0, a(0) = 1.
a(n) = 1 + 2 * A002457(n-1) = 1 + 2 * (2*n-1)!/(n-1)!^2 for n > 0, a(0) = 1.

A143567 E.g.f. satisfies A(x) = exp(x*A(x^3/3!)).

Original entry on oeis.org

1, 1, 1, 1, 5, 21, 61, 211, 1401, 8065, 37241, 240021, 1997821, 14657501, 105629525, 958412911, 9201199281, 86311594881, 871038486001, 9432024424585, 106531641929781, 1271523772132741, 15583607760968941, 194983864950339851
Offset: 0

Views

Author

Alois P. Heinz, Aug 24 2008, Aug 25 2008

Keywords

Links

Crossrefs

3rd column of A143565.
Cf. A367719.

Programs

  • Maple
    A:= proc(n) option remember; if n<=0 then 1 else unapply (convert (series (exp (x*A(n-3)(x^3/6)), x,n+1), polynom),x) fi end: a:= n-> coeff (A(n)(x), x,n)*n!: seq(a(n), n=0..29);
  • Mathematica
    A[n_] := A[n] = If[n <= 0, 1&, Function[Normal[Series[Exp[y*A[n-2][y^3/3!]], {y, 0, n+1}] /. y -> #]]]; a[n_] := Coefficient [A[n][x], x, n]*n!; Table[a[n], {n, 0, 29}] (* Jean-François Alcover, Feb 13 2014, after Maple *)

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=0..floor((n-1)/3)} (3*k+1) * a(k) * a(n-1-3*k) / (6^k * k! * (n-1-3*k)!). - Seiichi Manyama, Nov 28 2023

A143568 E.g.f. satisfies A(x) = exp(x*A(x^4/4!)).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 31, 106, 281, 946, 7561, 54286, 281161, 1207636, 7997991, 81996916, 701522641, 4580581916, 29742355441, 306369616636, 3632198902321, 34977922146721, 282526761829621, 2720464688299821, 36188717552636881, 464906756446099276, 4985291127563074901
Offset: 0

Views

Author

Alois P. Heinz, Aug 24 2008

Keywords

Links

Crossrefs

4th column of A143565.
Cf. A367720.

Programs

  • Maple
    A:= proc(n) option remember; if n<=0 then 1 else
      unapply(convert(series(exp(x*A(n-4)(x^4/24)), x, n+1), polynom), x) fi
    end:
a:= n-> coeff(A(n)(x), x,n)*n!:
seq(a(n), n=0..30);
  • Mathematica
    A[n_] := A[n] = If[n <= 0, 1&, Function[Normal[Series[Exp[y*A[n-2][y^4/4!]], {y, 0, n+1}] /. y -> #]]]; a[n_] := Coefficient[A[n][x], x, n]*n!; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 13 2014, after Maple *)

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=0..floor((n-1)/4)} (4*k+1) * a(k) * a(n-1-4*k) / (24^k * k! * (n-1-4*k)!). - Seiichi Manyama, Nov 28 2023

A143566 E.g.f. satisfies A(x) = exp(x*A(x^2/2!)).

Original entry on oeis.org

1, 1, 1, 4, 13, 46, 241, 1471, 9409, 67348, 564841, 4771801, 45459481, 463867834, 5060656693, 58878140686, 730612429681, 9556314730456, 131627520296929, 1912237000523623, 29032781640572881, 462811831018070206, 7687624300327129621, 133275225843052767244
Offset: 0

Views

Author

Alois P. Heinz, Aug 24 2008

Keywords

Links

Crossrefs

2nd column of A143565.
Cf. A138292.

Programs

  • Maple
    A:= proc(n) option remember; if n<=0 then 1 else unapply(convert(
       series(exp(x*A(n-2)(x^2/2)), x,n+1), polynom),x) fi
    end:
a:= n-> coeff(A(n)(x), x,n)*n!:
seq(a(n), n=0..28);
  • Mathematica
    A[n_] := A[n] = If[n <= 0, 1&, Function[Normal[Series[Exp[y*A[n-2][y^2/2]], {y, 0, n+1}] /. y -> #]]]; a[n_] := Coefficient[A[n][x], x, n]*n!; Table[a[n], {n, 0, 28}] (* Jean-François Alcover, Feb 13 2014, after Maple *)

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=0..floor((n-1)/2)} (2*k+1) * a(k) * a(n-1-2*k) / (2^k * k! * (n-1-2*k)!). - Seiichi Manyama, Nov 28 2023

A143569 E.g.f. satisfies A(x) = exp(x*A(x^5/5!)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 7, 43, 169, 505, 1261, 4159, 38809, 334621, 2036035, 9489481, 38390353, 257371297, 3131783929, 32230292725, 246760346161, 1493969858641, 9196517088991, 101815213853431, 1450104259874425, 16645720979718601, 147298665834676357
Offset: 0

Views

Author

Alois P. Heinz, Aug 24 2008

Keywords

Links

Crossrefs

5th column of A143565.

Programs

  • Maple
    A:= proc(n) option remember; if n<=0 then 1 else unapply (convert (series (exp (x*A(n-5)(x^5/120)), x,n+1), polynom),x) fi end: a:= n-> coeff (A(n)(x), x,n)*n!: seq(a(n), n=0..31);
  • Mathematica
    A[n_] := A[n] = If[n <= 0, 1&, Function[Normal[Series[Exp[y*A[n-2][y^5/5!]], {y, 0, n+1}] /. y -> #]]]; a[n_] := Coefficient[A[n][x], x, n]*n!; Table[a[n], {n, 0, 31}] (* Jean-François Alcover, Feb 13 2014, after Maple *)

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=0..floor((n-1)/5)} (5*k+1) * a(k) * a(n-1-5*k) / (120^k * k! * (n-1-5*k)!). - Seiichi Manyama, Nov 29 2023

A143570 E.g.f. satisfies A(x) = exp(x*A(x^6/6!)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 8, 57, 253, 841, 2311, 5545, 18019, 192193, 1936936, 13533521, 71607537, 308979217, 1195354525, 8070684721, 113661781381, 1368278263969, 12100291273456, 83294670263113, 474179436692501, 2787857745272601, 32561274444909211
Offset: 0

Views

Author

Alois P. Heinz, Aug 24 2008

Keywords

Links

Crossrefs

6th column of A143565.

Programs

  • Maple
    A:= proc(n) option remember; if n<=0 then 1 else unapply (convert (series (exp (x*A(n-6)(x^6/720)), x,n+1), polynom),x) fi end: a:= n-> coeff (A(n)(x), x,n)*n!: seq(a(n), n=0..33);
  • Mathematica
    A[n_] := A[n] = If[n <= 0, 1&, Function[Normal[Series[Exp[y*A[n-2][y^6/6!]], {y, 0, n+1}] /. y -> #]]]; a[n_] := Coefficient[A[n][x], x, n]*n!; Table[a[n], {n, 0, 33}] (* Jean-François Alcover, Feb 13 2014, after Maple *)

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=0..floor((n-1)/6)} (6*k+1) * a(k) * a(n-1-6*k) / (720^k * k! * (n-1-6*k)!). - Seiichi Manyama, Nov 29 2023

A143571 E.g.f. satisfies A(x) = exp(x*A(x^7/7!)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 9, 73, 361, 1321, 3961, 10297, 24025, 77221, 926641, 10696401, 84365425, 499445857, 2395445521, 9778915441, 36584246161, 248210675593, 3971313933049, 54773770095001, 549282704399001, 4258482133019401, 27025791550397641
Offset: 0

Views

Author

Alois P. Heinz, Aug 24 2008

Keywords

Links

Crossrefs

7th column of A143565.

Programs

  • Maple
    A:= proc(n) option remember; if n<=0 then 1 else unapply(convert(
      series(exp(x*A(n-7)(x^7/5040)), x, n+1), polynom), x) fi
    end:
a:= n-> coeff(A(n)(x), x,n)*n!:
seq(a(n), n=0..34);
  • Mathematica
    A[n_] := A[n] = If[n <= 0, 1&, Function[Normal[Series[Exp[y*A[n-2][y^7/7!]], {y, 0, n+1}] /. y -> #]]]; a[n_] := Coefficient[A[n][x], x, n]*n!; Table[a[n], {n, 0, 34}] (* Jean-François Alcover, Feb 13 2014, after Maple *)

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=0..floor((n-1)/7)} (7*k+1) * a(k) * a(n-1-7*k) / (5040^k * k! * (n-1-7*k)!). - Seiichi Manyama, Nov 29 2023

A143572 E.g.f. satisfies A(x) = exp(x*A(x^8/8!)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 91, 496, 1981, 6436, 18019, 45046, 102961, 328186, 4375801, 56951038, 500352841, 3276290746, 17289324361, 77309034166, 302908144177, 1104328093276, 7519851360451, 134741602227376, 2095457847783301, 23492070829121896
Offset: 0

Views

Author

Alois P. Heinz, Aug 24 2008

Keywords

Links

Crossrefs

8th column of A143565.

Programs

  • Maple
    A:= proc(n) option remember; if n<=0 then 1 else unapply (convert (series (exp (x*A(n-8)(x^8/40320)), x,n+1), polynom),x) fi end: a:= n-> coeff (A(n)(x), x,n)*n!: seq(a(n), n=0..35);
  • Mathematica
    A[n_] := A[n] = If[n <= 0, 1&, Function[Normal[Series[Exp[y*A[n-2][y^8/8!]], {y, 0, n+1}] /. y -> #]]]; a[n_] := Coefficient[A[n][x], x, n]*n!; Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Feb 13 2014, after Maple *)

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=0..floor((n-1)/8)} (8*k+1) * a(k) * a(n-1-8*k) / (40320^k * k! * (n-1-8*k)!). - Seiichi Manyama, Nov 29 2023

A143573 E.g.f. satisfies A(x) = exp(x*A(x^9/9!)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 111, 661, 2861, 10011, 30031, 80081, 194481, 437581, 1385671, 20323161, 294517861, 2851708861, 20461620411, 117812647921, 572637720601, 2430703053351, 9228958338601, 32965820988101, 225123959060001, 4466029537119151
Offset: 0

Views

Author

Alois P. Heinz, Aug 24 2008

Keywords

Links

Crossrefs

9th column of A143565.

Programs

  • Maple
    A:= proc(n) option remember; if n<=0 then 1 else unapply (convert (series (exp (x*A(n-9)(x^9/362880)), x,n+1), polynom),x) fi end: a:= n-> coeff (A(n)(x), x,n)*n!: seq(a(n), n=0..36);
  • Mathematica
    A[n_] := A[n] = If[n <= 0, 1&, Function[Normal[Series[Exp[y*A[n-2][y^9/9!]], {y, 0, n+1}] /. y -> #]]]; a[n_] := Coefficient[A[n][x], x, n]*n!; Table[a[n], {n, 0, 36}] (* Jean-François Alcover, Feb 13 2014, after Maple *)

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=0..floor((n-1)/9)} (9*k+1) * a(k) * a(n-1-9*k) / (362880^k * k! * (n-1-9*k)!). - Seiichi Manyama, Nov 29 2023
Showing 1-9 of 9 results.

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