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 1-6 of 6 results.

A100847 Number of partitions of 2n in which each odd part has even multiplicity and each even part has odd multiplicity.

Original entry on oeis.org

1, 2, 3, 7, 10, 17, 28, 42, 62, 93, 137, 193, 276, 383, 532, 734, 997, 1342, 1807, 2400, 3177, 4190, 5478, 7130, 9245, 11923, 15305, 19591, 24957, 31673, 40075, 50518, 63460, 79523, 99296, 123664, 153616, 190271, 235072, 289776, 356302, 437107, 535112, 653626
Offset: 0

Views

Author

Vladeta Jovovic, Aug 16 2007

Keywords

Examples

			a(3) = 7 because we have 6, 42, 411, 33, 222, 21111 and 111111.

Links

Crossrefs

Cf. A055922, A117958, A130126, A131942, A102247, A263401.

Programs

  • Maple
    g:=product((1+x^i-x^(2*i))/(1-x^i),i=1..50): gser:=series(g,x=0,40): seq(coeff(gser,x,n),n=0..35); # Emeric Deutsch, Aug 25 2007
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(`if`(irem(i+j, 2)=0, 0, b(n-i*j, i-1)), j=1..n/i)
       +b(n, i-1)))
    end:
a:= n-> b(2*n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, May 31 2014
  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1+x^k-x^(2*k))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 03 2016 *)

Formula

G.f.: Product_{i>0} (1+x^i-x^(2*i))/(1-x^i).
a(n) ~ sqrt(Pi^2/3 + 4*log(phi)^2) * exp(sqrt((2*Pi^2/3 + 8*log(phi)^2)*n)) / (4*Pi*n), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jan 03 2016

Extensions

More terms from Emeric Deutsch, Aug 25 2007

A102247 Number of partitions of n in which each odd part has odd multiplicity and each even part has even multiplicity.

Original entry on oeis.org

1, 1, 0, 2, 2, 3, 2, 4, 7, 8, 8, 10, 17, 17, 20, 26, 39, 39, 46, 56, 77, 85, 96, 116, 154, 172, 190, 234, 289, 328, 364, 440, 532, 610, 670, 808, 957, 1091, 1204, 1432, 1675, 1905, 2110, 2476, 2867, 3255, 3608, 4184, 4837, 5451, 6050, 6960, 7980, 8961, 9972, 11370
Offset: 0

Views

Author

Vladeta Jovovic, Aug 16 2007

Keywords

Examples

			a(7) = 4 because we have 7, 322, 22111 and 1111111.

Links

Crossrefs

Cf. A055922, A117958, A130126, A131942, A100847.

Programs

  • Maple
    g:=product((1+x^(2*i-1)-x^(4*i-2))/(1-x^(2*i)),i=1..40): gser:=series(g,x=0, 60): seq(coeff(gser,x,n),n=0..55); # Emeric Deutsch, Aug 23 2007
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(`if`(irem(i+j, 2)=0, b(n-i*j, i-1), 0), j=1..n/i)
       +b(n, i-1)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, May 31 2014
  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k-1) - x^(4*k-2))/(1-x^(2*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 03 2016 *)

Formula

G.f.: Product_{i>0} (1+x^(2*i-1)-x^(4*i-2))/(1-x^(2*i)).
a(n) ~ sqrt(Pi^2/3 + 4*log(phi)^2) * exp(sqrt((Pi^2/3 + 4*log(phi)^2)*n)) / (4*Pi*n), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jan 03 2016

Extensions

More terms from Emeric Deutsch, Aug 23 2007

A130127 Triangle defined by A000012 * A130125, read by rows.

Original entry on oeis.org

1, 1, 2, 2, 2, 4, 2, 4, 4, 8, 3, 4, 8, 8, 16, 3, 6, 8, 16, 16, 32, 4, 6, 12, 16, 32, 32, 64, 4, 8, 12, 24, 32, 64, 64, 128, 5, 8, 16, 24, 48, 64, 128, 128, 256, 5, 10, 16, 32, 48, 96, 128, 256, 256, 512, 6, 10, 20, 32, 64, 96, 192, 256, 512, 512, 1024, 6, 12, 20, 40, 64, 128, 192, 384, 512, 1024, 1024, 2048
Offset: 1

Views

Author

Gary W. Adamson, May 11 2007

Keywords

Comments

Row sums = A011377: (1, 3, 8, 18, 39, ...). A130126 = A130125 * A000012.

Examples

			First few rows of the triangle:
  1;
  1, 2;
  2, 2,  4;
  2, 4,  4,  8;
  3, 4,  8,  8, 16;
  3, 6,  8, 16, 16, 32;
  4, 6, 12, 16, 32, 32, 64;
  ...

Links

Crossrefs

Cf. A000012, A130125, A130126, A011377.

Programs

  • Magma
    [[2^(k-1)*Floor((n-k+2)/2): k in [1..n]]: n in [1..12]]; // G. C. Greubel, Jun 06 2019
  • Mathematica
    Table[2^(k-1)*Floor[(n-k+2)/2], {n,1,12}, {k,1,n}]//Flatten (* G. C. Greubel, Jun 06 2019 *)
  • PARI
    {T(n,k) = 2^(k-1)*floor((n-k+2)/2)}; \\ G. C. Greubel, Jun 06 2019
  • Sage
    [[2^(k-1)*floor((n-k+2)/2) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jun 06 2019

Formula

T(n,k) = 2^(k-1) * floor((n-k+2)/2). - G. C. Greubel, Jun 06 2019

Extensions

More terms added by G. C. Greubel, Jun 06 2019

A130275 Number of degree-n permutations such that number of cycles of size 2k is odd (or zero) for every k.

Original entry on oeis.org

1, 1, 2, 6, 21, 105, 675, 4725, 35805, 322245, 3236625, 35602875, 425872755, 5536345815, 77347084815, 1160206272225, 18403556596425, 312860462139225, 5643104418376425, 107218983949152075, 2136610763952639975, 44868826043005439475, 986129980012277775675
Offset: 0

Views

Author

Vladeta Jovovic, Aug 06 2007

Keywords

Examples

			a(4)=21 because only the following three degree-4 permutations do not qualify: (12)(34), (13)(24) and (14)(23).

Links

Crossrefs

Cf. A003483, A006950, A015128, A102759, A130126, A131942, A130219-A130223.

Programs

  • Maple
    g:=sqrt((1+x)/(1-x))*(product(1+sinh(x^(2*k)/(2*k)),k=1..30)): gser:=series(g, x=0,25): seq(factorial(n)*coeff(gser,x,n),n=0..20); # Emeric Deutsch, Aug 24 2007
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      `if`(j=0 or irem(i, 2)=1 or irem(j, 2)=1, multinomial(n,
       n-i*j, i$j)*(i-1)!^j/j!*b(n-i*j, i-1), 0), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 09 2015
  • Mathematica
    multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[If[j == 0 || Mod[i, 2] == 1 || Mod[j, 2] == 1, multinomial[n, Join[{n - i*j}, Array[i &, j]]]*(i - 1)!^j/j!*b[n - i*j, i - 1], 0], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Dec 22 2016, after Alois P. Heinz *)

Formula

E.g.f.: sqrt((1+x)/(1-x))*Product_{k>0} (1+sinh(x^(2*k)/(2*k))).

Extensions

More terms from Emeric Deutsch, Aug 24 2007

A130276 Number of degree-2n permutations such that number of cycles of size 2k-1 is even (or zero) for every k.

Original entry on oeis.org

1, 2, 16, 416, 20224, 1645312, 196388864, 33279311872, 7427338829824, 2151276556845056, 771086221948223488, 340572557390992900096, 179222835344084459061248, 112158801651454395931426816, 81399358513573250066141937664, 68530340884909785149816189222912
Offset: 0

Views

Author

Vladeta Jovovic, Aug 06 2007

Keywords

Examples

			a(2)=16 because there are 8 permutations that do not qualify: (1)(234), (1)(243), (123)(4), (124)(3), (132)(4), (134)(2), (142)(3) and (143)(2).

Links

Crossrefs

Cf. A003483, A006950, A015128, A102759, A130126, A131942, A130219-A130223.

Programs

  • Maple
    g:=(product(cosh(x^(2*k-1)/(2*k-1)),k=1..30))/sqrt(1-x^2): gser:=series(g,x= 0,30): seq(factorial(2*n)*coeff(gser,x,2*n),n=0..13); # Emeric Deutsch, Aug 24 2007
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      `if`(j=0 or irem(i, 2)=0 or irem(j, 2)=0, multinomial(n,
       n-i*j, i$j)*(i-1)!^j/j!*b(n-i*j, i-1), 0), j=0..n/i)))
    end:
a:= n-> b(2*n$2):
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 09 2015
  • Mathematica
    multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[If[j == 0 || Mod[i, 2] == 0 || Mod[j, 2] == 0, multinomial[n, Join[{n - i*j}, Array[i &, j]]]*(i - 1)!^j/j!*b[n - i*j, i - 1], 0], {j, 0, n/i}]]]; a[n_] := b[2n, 2n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Dec 22 2016, after Alois P. Heinz *)
  • PARI
    N=31; x='x+O('x^N);
v0=Vec(serlaplace(1/sqrt(1-x^2)*prod(k=1,N, cosh(x^(2*k-1)/(2*k-1)))));
vector(#v0\2,n,v0[2*n-1]) \\ Joerg Arndt, Jan 03 2011

Formula

E.g.f. with interleaved zeros: 1/sqrt(1-x^2)*Product_{k>=1} cosh(x^(2*k-1)/(2*k-1)). - Geoffrey Critzer, Jan 02 2011

Extensions

More terms from Emeric Deutsch, Aug 24 2007

A130278 Number of degree-n permutations such that number of cycles of size 2k-1 is odd (or zero) for every k.

Original entry on oeis.org

1, 1, 1, 6, 17, 100, 529, 3766, 31121, 276984, 2755553, 29665306, 364627801, 4639937380, 64952094401, 973467571350, 15750475301921, 264870218828656, 4759194994114369, 90124395399063730, 1812001488739061417, 37956199941196210716, 832297726351555617569
Offset: 0

Views

Author

Vladeta Jovovic, Aug 06 2007

Keywords

Examples

			a(4)=17 because only the following 7 permutations do not qualify: (1)(2)(3)(4), (1)(2)(34), (1)(23)(4), (1)(24)(3), (12)(3)(4), (13)(2)(4) and (14)(2)(3).

Links

Crossrefs

Cf. A003483, A006950, A015128, A102759, A130126, A131942, A130219-A130223.

Programs

  • Maple
    g:=(product(1+sinh(x^(2*k-1)/(2*k-1)),k=1..30))/sqrt(1-x^2): gser:=series(g,x =0,25): seq(factorial(n)*coeff(gser,x,n),n=0..20); # Emeric Deutsch, Aug 24 2007
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      `if`(j=0 or irem(i, 2)=0 or irem(j, 2)=1, multinomial(n,
       n-i*j, i$j)*(i-1)!^j/j!*b(n-i*j, i-1), 0), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 09 2015
  • Mathematica
    multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[If[j == 0 || Mod[i, 2] == 0 || Mod[j, 2] == 1, multinomial[n, Join[{n - i*j}, Array[i&, j]]]*(i - 1)!^j/j!*b[n - i*j, i - 1], 0], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Dec 22 2016, after Alois P. Heinz *)

Formula

E.g.f.: 1/sqrt(1-x^2)*Product_{k>0} (1+sinh(x^(2*k-1)/(2*k-1))).

Extensions

More terms from Emeric Deutsch, Aug 24 2007
Showing 1-6 of 6 results.

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