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.

Previous Showing 41-50 of 53 results. Next

A352254 Expansion of e.g.f. exp( x * sinh(x) / 2 ) (even powers only).

Original entry on oeis.org

1, 1, 5, 48, 753, 16880, 507579, 19509042, 927229553, 53126200872, 3597373129635, 283321938437318, 25614466939850169, 2629191169850594388, 303549146372282854883, 39103024746814973908890, 5581172267077778765676129, 877211696663645448333041072, 151002471269513108372760683523
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 09 2022

Keywords

Crossrefs

Cf. A000248, A000807, A003724, A003727, A005046, A009233, A351937, A352253.

Programs

  • Mathematica
    nmax = 36; Take[CoefficientList[Series[Exp[x Sinh[x]/2], {x, 0, nmax}], x] Range[0, nmax]!, {1, -1, 2}]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[2 n - 1, 2 k - 1] k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
  • PARI
    my(x='x+O('x^40), v=Vec(serlaplace(exp(x*sinh(x)/2)))); vector(#v\2, k, v[2*k-1]) \\ Michel Marcus, Mar 10 2022

Formula

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

A352624 Expansion of e.g.f. exp(exp(x) + cosh(x) - 2).

Original entry on oeis.org

1, 1, 3, 8, 31, 122, 579, 2886, 16139, 95358, 611111, 4128830, 29709695, 224400022, 1785322699, 14841968646, 129015458195, 1167021383902, 10979895178511, 107113768171950, 1082508179141031, 11308614423992102, 121995294474174963, 1356835055606851286, 15542964081299602811
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 24 2022

Keywords

Links

Crossrefs

Cf. A000110, A000807, A001861, A003724, A005046, A352327, A352617.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-k)*binomial(n-1, k-1)*(2-(k mod 2)), k=1..n))
    end:
seq(a(n), n=0..24);  # Alois P. Heinz, Mar 24 2022
  • Mathematica
    nmax = 24; CoefficientList[Series[Exp[Exp[x] + Cosh[x] - 2], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = (1/2) Sum[Binomial[n - 1, k - 1] (3 + (-1)^k) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 24}]

Formula

a(0) = 1; a(n) = (1/2) * Sum_{k=1..n} binomial(n-1,k-1) * (3 + (-1)^k) * a(n-k).
a(n) = Sum_{k=0..floor(n/2)} binomial(n,2*k) * A005046(k) * A000110(n-2*k).
a(n) = Sum_{k=0..floor(n/2)} binomial(n,2*k) * A000807(k) * A003724(n-2*k).

A361804 Number of partitions of [n] with an equal number of even and odd block sizes.

Original entry on oeis.org

1, 0, 0, 3, 0, 15, 45, 63, 1260, 1515, 25515, 104973, 510345, 5679765, 17252235, 263214318, 1207222380, 11863296915, 101718989235, 630468648873, 8281982665215, 48583038314415, 656006633919945, 5122900223419938, 54304561161840825, 605082149235374265
Offset: 0

Views

Author

Alois P. Heinz, Jun 12 2023

Keywords

Comments

Half the number of block sizes are even and the other half are odd.

Examples

			a(0) = 1: () the empty partition.
a(1) = 0.
a(2) = 0.
a(3) = 3: 12|3, 13|2, 1|23.
a(4) = 0.
a(5) = 15: 1234|5, 1235|4, 123|45, 1245|3, 124|35, 125|34, 12|345, 1345|2, 134|25, 135|24, 13|245, 145|23, 14|235, 15|234, 1|2345.
a(6) = 45: 12|34|5|6, 12|35|4|6, 12|3|45|6, 12|36|4|5, 12|3|46|5, 12|3|4|56, 13|24|5|6, 13|25|4|6, 13|2|45|6, 13|26|4|5, 13|2|46|5, 13|2|4|56, 14|23|5|6, 15|23|4|6, 1|23|45|6, 16|23|4|5, 1|23|46|5, 1|23|4|56, 14|25|3|6, 14|2|35|6, 14|26|3|5, 14|2|36|5, 14|2|3|56, 15|24|3|6, 1|24|35|6, 16|24|3|5, 1|24|36|5, 1|24|3|56, 15|2|34|6, 1|25|34|6, 16|2|34|5, 1|26|34|5, 1|2|34|56, 15|26|3|4, 15|2|36|4, 15|2|3|46, 16|25|3|4, 1|25|36|4, 1|25|3|46, 16|2|35|4, 1|26|35|4, 1|2|35|46, 16|2|3|45, 1|26|3|45, 1|2|36|45.

Links

Crossrefs

Cf. A000110, A003724, A004767, A004773, A005046, A275679.

Programs

  • Maple
    b:= proc(n, x, y) option remember; `if`(abs(x-y)>2*n, 0,
     `if`(n=0, 1, b(n-1, x+1, y)+`if`(x>0, b(n-1, x-1, y+1)*x, 0)+
     `if`(y>0, b(n-1, x+1, y-1)*y, 0)))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..33);

Formula

a(n) mod 5 = 3 for n in { A004767 }, a(n) mod 5 = 1 for n = 0 and a(n) mod 5 = 0 for all other n (n in { A004773 } \ { 0 }).
a(n) mod 3 = 0 for n >= 1.

A089004 Number of partitions of an n-element set that have at least one odd block.

Original entry on oeis.org

1, 1, 5, 11, 52, 172, 877, 3761, 21147, 109419, 678570, 4063248, 27644437, 186525861, 1382958545, 10323844183, 82864869804, 675378319788, 5832742205057, 51386368744773, 474869816156751, 4486977535640087
Offset: 1

Views

Author

Vladeta Jovovic, Nov 02 2003

Keywords

Links

Crossrefs

Cf. A000110, A005046, A003724, A086543, A087137.

Programs

  • Maple
    with(combinat):
b:= proc(n, i, t) option remember; `if`(n=0, t, `if`(i<1, 0,
       add(multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1,
       max(t, `if`(j=0, 0, irem(i, 2)))), j=0..n/i)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=1..30);  # Alois P. Heinz, Mar 08 2015
  • Mathematica
    With[{nn=30},CoefficientList[Series[Exp[Cosh[x]-1](Exp[Sinh[x]]-1),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 04 2018 *)

Formula

E.g.f.: exp(cosh(x)-1)*(exp(sinh(x))-1).

A089005 Number of partitions of n-set with at least one even block.

Original entry on oeis.org

0, 1, 3, 10, 40, 166, 749, 3683, 19275, 107806, 640970, 4024912, 26653653, 185401581, 1350624721, 10282222002, 81592209580, 673535269054, 5773214891137, 51291776763863, 471617190143567, 4481375500319334, 43947651280912186, 444258975094335440
Offset: 1

Views

Author

Vladeta Jovovic, Nov 02 2003

Keywords

Links

Crossrefs

Cf. A000110, A005046, A003724, A059838, A047967.

Programs

  • Maple
    with(combinat):
b:= proc(n, i, t) option remember; `if`(n=0, t, `if`(i<1, 0,
       add(multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1,
       max(t, `if`(j=0, 0, 1-irem(i, 2)))), j=0..n/i)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=1..30);
  • Mathematica
    multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, t_] := b[n, i, t] = If[n == 0, t, If[i<1, 0, Sum[multinomial[n, {n - i j} ~Join~ Table[i, {j}]]/j! b[n - i j, i - 1, Max[t, If[j == 0, 0, 1 - Mod[i, 2]]]], {j, 0, n/i}]]];
a[n_] := b[n, n, 0];
Array[a, 30] (* Jean-François Alcover, Nov 18 2020, after Maple *)
  • PARI
    my(x='x+O('x^30)); concat(0, Vec(serlaplace(exp(sinh(x))*(exp(cosh(x)-1)-1))))

Formula

E.g.f.: exp(sinh(x))*(exp(cosh(x)-1)-1).

A113774 Number of partitions of {1,...,n} into block sizes not a multiple of 3.

Original entry on oeis.org

1, 1, 2, 4, 11, 32, 112, 415, 1732, 7678, 37115, 190016, 1039546, 5996083, 36528196, 233492044, 1564012751, 10940385668, 79762304116, 604791685063, 4760047233424, 38825234812882, 327641201731475, 2856835856307428, 25702896025566886, 238331921722835203
Offset: 0

Views

Author

Vladeta Jovovic, Jan 19 2006

Keywords

Links

Crossrefs

Cf. A003724, A115276, A000246, A102736, A088009, A001590.

Programs

  • Maple
    nmax := 30: B := add(op(1+(i mod 3),[0,1,1])*x^i/i!,i=0..nmax) : egf := 0 : for i from 0 to nmax do egf := convert(egf+taylor(B^i,x=0,nmax+1)/i!,polynom) : od: for i from 0 to nmax do printf("%d ", i!*coeftayl(egf,x=0,i)) ; od: # R. J. Mathar, Feb 06 2008
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(`if`(
      irem(j, 3)=0, 0, binomial(n-1, j-1)*a(n-j)), j=1..n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 17 2015
  • Mathematica
    a=Sum[x^(3i)/(3i)!,{i,1,20}]; Range[0, 20]! CoefficientList[Series[Exp[Exp[x] - 1 - a], {x, 0, 20}], x] (* Geoffrey Critzer, Jan 02 2011 *)

Formula

E.g.f.: exp(B(x)), where B(x) is e.g.f. of A011655.

Extensions

More terms from R. J. Mathar, Feb 06 2008

A136631 Antidiagonal sums of triangle A136630, omitting antidiagonals of all zeros.

Original entry on oeis.org

1, 1, 2, 6, 28, 177, 1449, 14869, 185230, 2738962, 47287352, 939759621, 21241309681, 540698975061, 15370957337418, 484433735633218, 16817886069720724, 639545680226171989, 26507567678760284105
Offset: 0

Views

Author

Paul D. Hanna, Jan 14 2008

Keywords

Crossrefs

Cf. A136630, A003724 (row sums of A136630).

Programs

  • PARI
    {a(n)=sum(k=0,n,polcoeff(1/prod(j=0,k\2,1-(2*j+(k%2))^2*x^2 +x*O(x^(2*n-2*k))),2*n-2*k))}

Formula

a(n) = Sum_{k=0..n} [x^(2n-2k)] Product_{j=0..[k/2]} 1/(1 - (2j + k-2[k/2])^2*x^2).

A352469 a(0) = 1; a(n) = (1/n) * Sum_{k=0..floor((n-1)/2)} binomial(n,2*k+1)^3 * (2*k+1) * a(n-2*k-1).

Original entry on oeis.org

1, 1, 4, 37, 640, 18401, 810616, 51506645, 4512303104, 526359723265, 79484297525704, 15182084413118525, 3598056798827450752, 1040872295660542894433, 362422517793599461361216, 150047916077302216370174237, 73081847594180657956494147584, 41481744863993143666887680079873
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 17 2022

Keywords

Crossrefs

Cf. A003724, A061684, A346220, A352466.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[n, 2 k + 1]^3 (2 k + 1) a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 17}]
nmax = 17; CoefficientList[Series[Exp[Sum[x^(2 k + 1)/(2 k + 1)!^3, {k, 0, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!^3

Formula

Sum_{n>=0} a(n) * x^n / n!^3 = exp( Sum_{n>=0} x^(2*n+1) / (2*n+1)!^3 ).

A363073 Number of set partitions of [n] such that each element is contained in a block whose block size parity coincides with the parity of the element.

Original entry on oeis.org

1, 1, 0, 0, 1, 2, 0, 0, 20, 48, 0, 0, 1147, 3968, 0, 0, 173203, 709488, 0, 0, 53555964, 246505600, 0, 0, 28368601065, 148963383616, 0, 0, 24044155851601, 141410718244864, 0, 0, 30934515698084780, 198914201874983936, 0, 0, 57215369885233295955, 398742900995358584320
Offset: 0

Views

Author

Alois P. Heinz, May 17 2023

Keywords

Comments

All odd elements are in blocks with an odd block size and all even elements are in blocks with an even block size.

Examples

			a(0) = 1: (), the empty partition.
a(1) = 1: 1.
a(4) = 1: 1|24|3.
a(5) = 2: 135|24, 1|24|3|5.
a(8) = 20: 135|2468|7, 135|24|68|7, 137|2468|5, 137|24|5|68, 135|26|48|7, 135|28|46|7, 137|26|48|5, 137|28|46|5, 157|2468|3, 157|24|3|68, 1|2468|357, 1|24|357|68, 1|2468|3|5|7, 1|24|3|5|68|7, 157|26|3|48, 157|28|3|46, 1|26|357|48, 1|28|357|46, 1|26|3|48|5|7, 1|28|3|46|5|7.

Links

Crossrefs

Cf. A000110, A003724, A005046, A124419, A274538, A275679.

Programs

  • Maple
    b:= proc(n, t) option remember; `if`(n=0, 1, add(
     `if`((j+t)::even, b(n-j, t)*binomial(n-1, j-1), 0), j=1..n))
    end:
a:= n-> (h-> b(n-h, 1)*b(h, 0))(iquo(n, 2)):
seq(a(n), n=0..40);
  • Mathematica
    b[n_, t_] := b[n, t] = If[n == 0, 1, Sum[If[EvenQ[j + t], b[n - j, t]* Binomial[n - 1, j - 1], 0], {j, 1, n}]];
a[n_] := b[n - #, 1]*b[#, 0]&[Quotient[n, 2]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Nov 18 2023, after Alois P. Heinz *)

Formula

a(n) = A003724(ceiling(n/2)) * A005046(floor(n/4)) if (n mod 4) in {0,1}.
a(n) = 0 if (n mod 4) in {2,3}.

A381147 E.g.f. A(x) satisfies A(x) = exp( sinh(x * A(x)) / A(x) ).

Original entry on oeis.org

1, 1, 1, 2, 13, 92, 621, 5112, 56057, 705168, 9480665, 141039648, 2366242693, 43609330624, 864164283269, 18414385180544, 422574196387953, 10374625080684800, 270563138370828465, 7472794772378583552, 218190569313134267517, 6714970997524417977344
Offset: 0

Views

Author

Seiichi Manyama, Feb 15 2025

Keywords

Crossrefs

Cf. A003724, A162650, A219503.
Cf. A136630.

Programs

  • PARI
    a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a(n) = sum(k=0, n, (n-k+1)^(k-1)*a136630(n, k));

Formula

a(n) = Sum_{k=0..n} (n-k+1)^(k-1) * A136630(n,k).
Previous Showing 41-50 of 53 results. Next

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

Content is available under The OEIS End-User License Agreement.