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 12 results. Next

A143406 Number of forests of labeled rooted trees of height at most 1, with n labels, where each root contains a nonempty set of labels of equal size, also row sums of A143398.

Original entry on oeis.org

1, 1, 4, 14, 55, 252, 1319, 7737, 50040, 351636, 2659375, 21519027, 185279186, 1688183135, 16206401020, 163376811610, 1724624368377, 19011582728772, 218312877627483, 2605840967052663, 32271957793959066, 413991491885677105, 5492584623675060620
Offset: 0

Views

Author

Alois P. Heinz, Aug 12 2008

Keywords

Examples

			a(2) = 4, because 4 forests with 2 labels exist: {1}{2}, {1}<-2, {2}<-1, {1,2}.

Links

Crossrefs

Cf. A143398, A000142.

Programs

  • Maple
    a:= n-> if n=0 then 1 else n! * add(add(i^(n-k*i)/
        ((n-k*i)!*i!*k!^i), i=1..floor(n/k)), k=1..n) fi:
seq(a(n), n=0..30);

Formula

a(n) = 1 if n=0 and a(n) = n! * Sum_{k=1..n} Sum_{i=1..floor(n/k)} i^(n-k*i)/ ((n-k*i)!*i!*k!^i) else.

A145460 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where sequence a_k of column k is the exponential transform of C(n,k).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 0, 3, 5, 1, 0, 1, 10, 15, 1, 0, 0, 3, 41, 52, 1, 0, 0, 1, 9, 196, 203, 1, 0, 0, 0, 4, 40, 1057, 877, 1, 0, 0, 0, 1, 10, 210, 6322, 4140, 1, 0, 0, 0, 0, 5, 30, 1176, 41393, 21147, 1, 0, 0, 0, 0, 1, 15, 175, 7273, 293608, 115975, 1, 0, 0, 0, 0, 0, 6, 35, 1176, 49932, 2237921, 678570
Offset: 0

Views

Author

Alois P. Heinz, Oct 10 2008

Keywords

Comments

A(n,k) is also the number of ways of placing n labeled balls into indistinguishable boxes, where in each filled box k balls are seen at the top. E.g. A(3,1)=10:
|1.| |2.| |3.| |1|2| |1|2| |1|3| |1|3| |2|3| |2|3| |1|2|3|
|23| |13| |12| |3|.| |.|3| |2|.| |.|2| |1|.| |.|1| |.|.|.|
+--+ +--+ +--+ +-+-+ +-+-+ +-+-+ +-+-+ +-+-+ +-+-+ +-+-+-+

Examples

			Square array A(n,k) begins:
   1,   1,  1,  1,  1,  1,  ...
   1,   1,  0,  0,  0,  0,  ...
   2,   3,  1,  0,  0,  0,  ...
   5,  10,  3,  1,  0,  0,  ...
  15,  41,  9,  4,  1,  0,  ...
  52, 196, 40, 10,  5,  1,  ...

Links

Crossrefs

Columns k=0-9 give: A000110, A000248, A133189, A145453, A145454, A145455, A145456, A145457, A145458, A145459.
A(2n,n) gives A029651.
Cf.: A007318, A143398, A292948.

Programs

  • Maple
    exptr:= proc(p) local g; g:=
          proc(n) option remember; `if`(n=0, 1,
             add(binomial(n-1, j-1) *p(j) *g(n-j), j=1..n))
        end: end:
A:= (n,k)-> exptr(i-> binomial(i, k))(n):
seq(seq(A(n, d-n), n=0..d), d=0..12);
  • Mathematica
    Exptr[p_] := Module[{g}, g[n_] := g[n] = If[n == 0, 1, Sum[Binomial[n-1, j-1] *p[j]*g[n-j], {j, 1, n}]]; g]; A[n_, k_] := Exptr[Function[i, Binomial[i, k]]][n]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Jan 15 2014, translated from Maple *)
  • Ruby
    def ncr(n, r)
  return 1 if r == 0
  (n - r + 1..n).inject(:*) / (1..r).inject(:*)
end
def A(k, n)
  ary = [1]
  (1..n).each{|i| ary << (0..i - 1).inject(0){|s, j| s + ncr(i - 1, j) * ncr(j + 1, k) * ary[i - 1 - j]}}
  ary
end
def A145460(n)
  a = []
  (0..n).each{|i| a << A(i, n - i)}
  ary = []
  (0..n).each{|i|
    (0..i).each{|j|
      ary << a[i - j][j]
    }
  }
  ary
end
p A145460(20) # Seiichi Manyama, Sep 28 2017

Formula

A(0,k) = 1 and A(n,k) = Sum_{i=0..n-1} binomial(n-1,i) * binomial(i+1,k) * A(n-1-i,k) for n > 0. - Seiichi Manyama, Sep 28 2017

A029651 Central elements of the (1,2)-Pascal triangle A029635.

Original entry on oeis.org

1, 3, 9, 30, 105, 378, 1386, 5148, 19305, 72930, 277134, 1058148, 4056234, 15600900, 60174900, 232676280, 901620585, 3500409330, 13612702950, 53017895700, 206769793230, 807386811660, 3156148445580, 12350146091400, 48371405524650
Offset: 0

Views

Author

Mohammad K. Azarian

Keywords

Comments

If Y is a fixed 2-subset of a (2n+1)-set X then a(n) is the number of (n+1)-subsets of X intersecting Y. - Milan Janjic, Oct 28 2007

References

  • V. N. Smith and L. Shapiro, Catalan numbers, Pascal's triangle and mutators, Congressus Numerant., 205 (2010), 187-197.

Links

Crossrefs

Cf. A001700, A143398, A145460.
Essentially a duplicate of A003409.

Programs

  • Maple
    a := n -> (3/2)*4^n*GAMMA(1/2+n)/(sqrt(Pi)*GAMMA(1+n))-0^n/2;
seq(simplify(a(n)), n=0..24); # Peter Luschny, Dec 16 2015
  • Mathematica
    Join[{1},Table[3*Binomial[2n-1,n],{n,30}]] (* Harvey P. Dale, Aug 11 2015 *)
  • PARI
    concat([1], for(n=1, 50, print1(3*binomial(2*n-1,n), ", "))) \\ G. C. Greubel, Jan 23 2017

Formula

a(n) = 3 * binomial(2n-1, n) (n>0). - Len Smiley, Nov 03 2001
a(n) = 3*A001700(n-1), (n>=1).
G.f.: (1+xC(x))/(1-2xC(x)), C(x) the g.f. of A000108. - Paul Barry, Dec 17 2004
a(n) = A003409(n), n>0. - R. J. Mathar, Oct 23 2008
a(n) = Sum_{k=0..n} A039599(n,k)*A000034(k). - Philippe Deléham, Oct 29 2008
a(n) = (3/2)*4^n*Gamma(1/2+n)/(sqrt(Pi)*Gamma(1+n))-0^n/2. - Peter Luschny, Dec 16 2015
a(n) ~ (3/2)*4^n*(1-(1/8)/n+(1/128)/n^2+(5/1024)/n^3-(21/32768)/n^4)/sqrt(n*Pi). - Peter Luschny, Dec 16 2015
a(n) = 2^(1-n)*Sum_{k=0..n} binomial(k+n,k)*binomial(2*n-1,n-k), n>0, a(0)=1. - Vladimir Kruchinin, Nov 23 2016
E.g.f.: (3*exp(2*x)*BesselI(0,2*x) - 1)/2. - Ilya Gutkovskiy, Nov 23 2016
a(n) = A143398(2n,n) = A145460(2n,n). - Alois P. Heinz, Sep 09 2018
a(n) = [x^n] C(-x)^(-3*n), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. - Peter Bala, Oct 16 2024

Extensions

More terms from David W. Wilson

A145453 Exponential transform of binomial(n,3) = A000292(n-2).

Original entry on oeis.org

1, 0, 0, 1, 4, 10, 30, 175, 1176, 7084, 42120, 286605, 2270180, 19213766, 166326524, 1497096055, 14374680880, 147259920760, 1582837679056, 17659771122969, 204674606377140, 2473357218561250, 31148510170120420, 407154732691440811, 5504706823227724904
Offset: 0

Views

Author

Alois P. Heinz, Oct 10 2008

Keywords

Comments

a(n) is the number of ways of placing n labeled balls into indistinguishable boxes, where in each filled box 3 balls are seen at the top.
a(n) is also the number of forests of labeled rooted trees of height at most 1, with n labels, where each root contains 3 labels.

Links

Crossrefs

3rd column of A145460, A143398.
Cf. A292889, A292910.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1,
      add(binomial(n-1, j-1) *binomial(j,3) *a(n-j), j=1..n))
    end:
seq(a(n), n=0..30);
  • Mathematica
    Table[Sum[BellY[n, k, Binomial[Range[n], 3]], {k, 0, n}], {n, 0, 25}] (* Vladimir Reshetnikov, Nov 09 2016 *)

Formula

E.g.f.: exp(exp(x)*x^3/3!).

A133189 Number of simple directed graphs on n labeled nodes consisting only of some cycle graphs C_2 and nodes not part of a cycle having directed edges to both nodes in exactly one cycle.

Original entry on oeis.org

1, 0, 1, 3, 9, 40, 210, 1176, 7273, 49932, 372060, 2971540, 25359411, 230364498, 2215550428, 22460391240, 239236043985, 2669869110856, 31134833803728, 378485082644400, 4786085290280275, 62838103267148790, 855122923978737876, 12042364529117844328
Offset: 0

Views

Author

Alois P. Heinz, Dec 17 2007

Keywords

Examples

			a(3) = 3, because there are 3 graphs of the given kind for 3 labeled nodes: 3->1<->2<-3,  2->1<->3<-2,  1->2<->3<-1.

References

  • A. P. Heinz (1990). Analyse der Grenzen und Möglichkeiten schneller Tableauoptimierung. PhD Thesis, Albert-Ludwigs-Universität Freiburg, Freiburg i. Br., Germany.

Links

Crossrefs

Cf. A006882, A007318, A135458, A135429.
2nd column of A145460, A143398.

Programs

  • Maple
    a:= proc(n) option remember; add(binomial(n, k+k)*
      doublefactorial(k+k-1) *k^(n-k-k), k=0..floor(n/2))
    end:
seq(a(n), n=0..30);
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(
      binomial(n-1, j-1) *binomial(j, 2) *a(n-j), j=1..n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 16 2015
  • Mathematica
    nn=20;Range[0,nn]!CoefficientList[Series[Exp[Exp[x]x^2/2],{x,0,nn}],x]  (* Geoffrey Critzer, Nov 23 2012 *)
Table[Sum[BellY[n, k, Binomial[Range[n], 2]], {k, 0, n}], {n, 0, 25}] (* Vladimir Reshetnikov, Nov 09 2016 *)

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(n,2*k) * A006882(2*k-1) * k^(n-2*k).
E.g.f.: exp(exp(x)*x^2/2). - Geoffrey Critzer, Nov 23 2012

A145454 Exponential transform of binomial(n,4) = A000332.

Original entry on oeis.org

1, 0, 0, 0, 1, 5, 15, 35, 105, 756, 6510, 46530, 283470, 1667380, 11457446, 99776040, 969295145, 9298091180, 86154691680, 804769174536, 8052676029420, 88489327173660, 1038440150703340, 12501684521410700, 151866259113256611
Offset: 0

Views

Author

Alois P. Heinz, Oct 10 2008

Keywords

Comments

a(n) is the number of ways of placing n labeled balls into indistinguishable boxes, where in each filled box 4 balls are seen at the top.
a(n) is also the number of forests of labeled rooted trees of height at most 1, with n labels, where each root contains 4 labels.

Links

Crossrefs

4th column of A145460, A143398.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(
      binomial(n-1, j-1) *binomial(j,4) *a(n-j), j=1..n))
    end:
seq(a(n), n=0..30);
  • Mathematica
    Table[Sum[BellY[n, k, Binomial[Range[n], 4]], {k, 0, n}], {n, 0, 25}] (* Vladimir Reshetnikov, Nov 09 2016 *)

Formula

E.g.f.: exp(exp(x)*x^4/4!).

A351703 Square array T(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 - x^k * exp(x) / k!).

Original entry on oeis.org

1, 1, 1, 1, 0, 4, 1, 0, 1, 21, 1, 0, 0, 3, 148, 1, 0, 0, 1, 12, 1305, 1, 0, 0, 0, 4, 70, 13806, 1, 0, 0, 0, 1, 10, 465, 170401, 1, 0, 0, 0, 0, 5, 40, 3591, 2403640, 1, 0, 0, 0, 0, 1, 15, 315, 31948, 38143377, 1, 0, 0, 0, 0, 0, 6, 35, 2296, 319068, 672552730
Offset: 0

Views

Author

Seiichi Manyama, Feb 20 2022

Keywords

Examples

			Square array begins:
      1,   1,  1,  1, 1, 1, ...
      1,   0,  0,  0, 0, 0, ...
      4,   1,  0,  0, 0, 0, ...
     21,   3,  1,  0, 0, 0, ...
    148,  12,  4,  1, 0, 0, ...
   1305,  70, 10,  5, 1, 0, ...
  13806, 465, 40, 15, 6, 1, ...

Crossrefs

Column k=1..5 gives A006153, A346888, A346889, A346890, A346893.
Cf. A143398, A351761.

Programs

  • PARI
    T(n, k) = if(n==0, 1, binomial(n, k)*sum(j=0, n-k, binomial(n-k, j)*T(j, k)));
  • PARI
    T(n, k) = n!*sum(j=0, n\k, j^(n-k*j)/(k!^j*(n-k*j)!)); \\ Seiichi Manyama, May 13 2022

Formula

T(0,k) = 1 and T(n,k) = binomial(n,k) * Sum_{j=0..n-k} binomial(n-k,j) * T(j,k) for n > 0.
T(n,k) = n! * Sum_{j=0..floor(n/k)} j^(n-k*j)/(k!^j * (n-k*j)!). - Seiichi Manyama, May 13 2022

A145455 Exponential transform of C(n,5) = A000389.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 6, 21, 56, 126, 378, 3234, 34056, 289575, 2020018, 12237225, 70634928, 468041756, 4190274648, 45557515620, 499503011496, 5121432107757, 49183015347774, 462331794763069, 4579813478536296, 50959878972009546
Offset: 0

Views

Author

Alois P. Heinz, Oct 10 2008

Keywords

Comments

a(n) is the number of ways of placing n labeled balls into indistinguishable boxes, where in each filled box 5 balls are seen at the top.
a(n) is also the number of forests of labeled rooted trees of height at most 1, with n labels, where each root contains 5 labels.

Links

Crossrefs

5th column of A145460, A143398.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1,
      add(binomial(n-1, j-1) *binomial(j,5) *a(n-j), j=1..n))
    end:
seq(a(n), n=0..30);
  • Mathematica
    With[{nmax = 50}, CoefficientList[Series[Exp[Exp[x]*x^5/5!], {x, 0, nmax}], x]] (* G. C. Greubel, Nov 21 2017 *)
  • PARI
    x='x+O('x^30); Vec(serlaplace(exp(exp(x)*x^5/5!))) \\ G. C. Greubel, Nov 21 2017

Formula

E.g.f.: exp(exp(x)*x^5/5!).

A145456 Exponential transform of C(n,6) = A000579.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 7, 28, 84, 210, 462, 1386, 13728, 171171, 1686685, 13461448, 91495768, 551777772, 3142726692, 19787406360, 172188951144, 1999835600301, 24655331721867, 285725747201356, 3034790658153100, 29876851476502030
Offset: 0

Views

Author

Alois P. Heinz, Oct 10 2008

Keywords

Comments

a(n) is the number of ways of placing n labeled balls into indistinguishable boxes, where in each filled box 6 balls are seen at the top.
a(n) is also the number of forests of labeled rooted trees of height at most 1, with n labels, where each root contains 6 labels.

Links

Crossrefs

6th column of A145460, A143398.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1,
      add(binomial(n-1, j-1) *binomial(j,6) *a(n-j), j=1..n))
    end:
seq(a(n), n=0..30);
  • Mathematica
    With[{nn=30},CoefficientList[Series[Exp[Exp[x] x^6/6!],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Apr 14 2018 *)

Formula

E.g.f.: exp(exp(x)*x^6/6!).

A145457 Exponential transform of C(n,7) = A000580.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 8, 36, 120, 330, 792, 1716, 5148, 57915, 835120, 9354488, 84047184, 638567124, 4256855760, 25607297880, 144863655024, 869425029957, 7081044528888, 83816629147900, 1131047706331400, 14634713798592030, 173380501913172840
Offset: 0

Views

Author

Alois P. Heinz, Oct 10 2008

Keywords

Comments

a(n) is the number of ways of placing n labeled balls into indistinguishable boxes, where in each filled box 7 balls are seen at the top.
a(n) is also the number of forests of labeled rooted trees of height at most 1, with n labels, where each root contains 7 labels.

Links

Crossrefs

7th column of A145460, A143398.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1,
      add(binomial(n-1, j-1) *binomial(j,7) *a(n-j), j=1..n))
    end:
seq(a(n), n=0..30);
  • Mathematica
    a[n_] := a[n] = If[n == 0, 1, Sum[Binomial[n - 1, j - 1] *Binomial[j, 7]* a[n - j], {j, 1, n}]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 11 2018, after Alois P. Heinz *)

Formula

E.g.f.: exp(exp(x)*x^7/7!).
Showing 1-10 of 12 results. Next

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