A102866 -id:A102866 - cp's OEIS frontend

A360634 Number T(n,k) of sets of nonempty words over binary alphabet with a total of n letters of which k are the first letter; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 1, 1, 1, 3, 1, 2, 6, 6, 2, 2, 11, 16, 11, 2, 3, 18, 37, 37, 18, 3, 4, 28, 73, 100, 73, 28, 4, 5, 42, 133, 228, 228, 133, 42, 5, 6, 61, 227, 470, 593, 470, 227, 61, 6, 8, 86, 370, 899, 1370, 1370, 899, 370, 86, 8, 10, 119, 580, 1617, 2894, 3497, 2894, 1617, 580, 119, 10

Offset: 0

Alois P. Heinz, Feb 14 2023

			T(4,0) = 2: {bbbb}, {b,bbb}.
T(4,1) = 11: {abbb}, {babb}, {bbab}, {bbba}, {a,bbb}, {ab,bb}, {abb,b}, {b,bab}, {b,bba}, {ba,bb}, {a,b,bb}.
T(4,2) = 16: {aabb}, {abab}, {abba}, {baab}, {baba}, {bbaa}, {a,abb}, {a,bab}, {a,bba}, {aa,bb}, {aab,b}, {ab,ba}, {aba,b}, {b,baa}, {a,ab,b}, {a,b,ba}.
Triangle T(n,k) begins:
   1;
   1,   1;
   1,   3,   1;
   2,   6,   6,    2;
   2,  11,  16,   11,    2;
   3,  18,  37,   37,   18,    3;
   4,  28,  73,  100,   73,   28,    4;
   5,  42, 133,  228,  228,  133,   42,    5;
   6,  61, 227,  470,  593,  470,  227,   61,   6;
   8,  86, 370,  899, 1370, 1370,  899,  370,  86,   8;
  10, 119, 580, 1617, 2894, 3497, 2894, 1617, 580, 119, 10;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Columns k=0-2 give: A000009, A095944, A360650.
Row sums give A102866.
T(2n,n) gives A360638.
Cf. A055375 (the same for multisets), A200751, A208741.

g:= proc(n, i, j) option remember; expand(`if`(j=0, 1, `if`(i<0, 0, add(
      g(n, i-1, j-k)*x^(i*k)*binomial(binomial(n, i), k), k=0..j))))
    end:
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
     `if`(i<1, 0, add(b(n-i*j, i-1)*g(i$2, j), j=0..n/i))))
    end:
T:= (n, k)-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..15);

g[n_, i_, j_] := g[n, i, j] = Expand[If[j == 0, 1, If[i < 0, 0, Sum[g[n, i - 1, j - k]*x^(i*k)*Binomial[Binomial[n, i], k], {k, 0, j}]]]];
b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1]*g[i, i, j], {j, 0, n/i}]]]];
T[n_] := CoefficientList[b[n, n], x];
Table[T[n], {n, 0, 15}] // Flatten (* Jean-François Alcover, Dec 05 2023, after Alois P. Heinz *)

T(n,k) = T(n,n-k).

Sum_{k=0..2n} (-1)^k*T(2n,k) = A200751(n). - Alois P. Heinz, Sep 09 2023

A261053 Expansion of Product_{k>=1} (1+x^k)^(k^k).

Original entry on oeis.org

1, 1, 4, 31, 289, 3495, 51268, 891152, 17926913, 409907600, 10499834497, 297793199060, 9262502810645, 313457634240463, 11464902463397642, 450646709610954343, 18943070964019019671, 847932498252050293971, 40266255926484893366914, 2021845081107882645459639

Offset: 0

Vaclav Kotesovec, Aug 08 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..385

Cf. A023880, A261052, A026007, A027998, A248882, A102866, A256142.

m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1+x^k)^(k^k): k in [1..(m+2)]]))); // G. C. Greubel, Nov 08 2018

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(i^i, j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 08 2015

nmax=20; CoefficientList[Series[Product[(1+x^k)^(k^k),{k,1,nmax}],{x,0,nmax}],x]

m=20; x='x+O('x^m); Vec(prod(k=1,m, (1+x^k)^(k^k))) \\ G. C. Greubel, Nov 08 2018

a(n) ~ n^n * (1 + exp(-1)/n + (exp(-1)/2 + 4*exp(-2))/n^2).

G.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d+1)*d^(d+1) ) * x^k/k). - Ilya Gutkovskiy, Nov 08 2018

A261052 Expansion of Product_{k>=1} (1+x^k)^(k!).

Original entry on oeis.org

1, 1, 2, 8, 31, 157, 915, 6213, 48240, 423398, 4147775, 44882107, 531564195, 6837784087, 94909482330, 1413561537884, 22482554909451, 380269771734265, 6815003300096013, 128992737080703803, 2571218642722865352, 53835084737513866662, 1181222084520177393143

Offset: 0

Vaclav Kotesovec, Aug 08 2015

nonn

Weigh transform of the factorial numbers. - Alois P. Heinz, Jun 11 2018

Alois P. Heinz, Table of n, a(n) for n = 0..450

Cf. A000142, A107895, A168246, A261053, A026007, A027998, A248882, A102866, A256142.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(i!, j)*b(n-i*j,i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 08 2015

nmax=25; CoefficientList[Series[Product[(1+x^k)^(k!),{k,1,nmax}],{x,0,nmax}],x]

seq(n)={Vec(exp(x*Ser(dirmul(vector(n, n, n!), -vector(n, n, (-1)^n/n)))))} \\ Andrew Howroyd, Jun 22 2018

a(n) ~ n! * (1 + 1/n + 2/n^2 + 10/n^3 + 57/n^4 + 401/n^5 + 3382/n^6 + 33183/n^7 + 371600/n^8 + 4685547/n^9 + 65792453/n^10).

A261051 Expansion of Product_{k>=1} (1+x^k)^(Lucas(k)).

Original entry on oeis.org

1, 1, 3, 7, 14, 33, 69, 148, 307, 642, 1314, 2684, 5432, 10924, 21841, 43431, 85913, 169170, 331675, 647601, 1259737, 2441706, 4716874, 9083215, 17439308, 33387589, 63749174, 121409236, 230658963, 437198116, 826838637, 1560410267, 2938808875, 5524005110

Offset: 0

Vaclav Kotesovec, Aug 08 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Asymptotics of the Euler transform of Fibonacci numbers, arXiv:1508.01796 [math.CO], Aug 07 2015

Cf. A000032, A261031, A261050, A026007, A027998, A248882, A102866, A256142.

L:= n-> (<<0|1>, <1|1>>^n. <<2, 1>>)[1, 1]:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       add(binomial(L(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Aug 08 2015

nmax=40; CoefficientList[Series[Product[(1+x^k)^LucasL[k],{k,1,nmax}],{x,0,nmax}],x]

a(n) ~ phi^n / (2*sqrt(Pi)*n^(3/4)) * exp(-1 + 1/(2*sqrt(5)) + 2*sqrt(n) + s), where s = Sum_{k>=2} (-1)^(k+1) * (2 + phi^k)/((phi^(2*k) - phi^k - 1)*k) = -0.590290697526802161885355317939144642488927381134222996704542... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio.

G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k*(1 + 2*x^k)/(k*(1 - x^k - x^(2*k)))). - Ilya Gutkovskiy, May 30 2018

A216158 The total number of nonempty words in all length n finite languages on an alphabet of two letters.

Original entry on oeis.org

0, 2, 6, 24, 72, 220, 652, 1848, 5160, 14130, 38102, 101296, 266328, 692740, 1785524, 4563888, 11577888, 29170128, 73032808, 181793136, 450100760, 1108868820, 2719167020, 6639085968, 16144137800, 39107596850, 94393612782, 227062741160, 544439640328, 1301446217244

Offset: 0

Geoffrey Critzer, Sep 03 2012

nonn

A finite language is a set of distinct words with size being the total number of letters in all words.

			a(3) = 24 because the sets (languages) are {a,aa}; {a,ab}; {a,ba}; {a,bb}; {b,aa}; {b,ab}; {b,ba}; {b,bb}; {aaa}; {aab}; {aba}; {abb}; {baa}; {bab}; {bba}; {bbb} where the distinct words are separated by commas.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 64.

Cf. A102866.

h:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, 0, add(
      (p-> p+[0, p[1]*j])(binomial(2^i, j)*h(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> h(n$2)[2]:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 24 2017

nn=30;p=Product[(1+y x^i)^(2^i),{i,1,nn}];CoefficientList[Series[D[p,y]/.y->1,{x,0,nn}],x]

a(n) = Sum_{k>0} k * A208741(n,k).

A319919 Expansion of Product_{k>=1} (1 + x^k)^(2^k-1).

Original entry on oeis.org

1, 1, 3, 10, 25, 70, 182, 476, 1220, 3122, 7883, 19794, 49340, 122237, 301114, 737923, 1799597, 4369204, 10563800, 25441377, 61048713, 145988775, 347981713, 826921992, 1959363778, 4629903905, 10911757432, 25652950459, 60165831361, 140792215037, 328750398275, 766041930160, 1781452975346

Offset: 0

Ilya Gutkovskiy, Oct 01 2018

nonn

Convolution of A081362 and A102866.

Weigh transform of A000225.

Vaclav Kotesovec, Table of n, a(n) for n = 0..3000
N. J. A. Sloane, Transforms

Cf. A000225, A007070, A081362, A098407, A102866, A319918.

a:=series(mul((1+x^k)^(2^k-1),k=1..100),x=0,33): seq(coeff(a,x,n),n=0..32); # Paolo P. Lava, Apr 02 2019

nmax = 32; CoefficientList[Series[Product[(1 + x^k)^(2^k - 1), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 32; CoefficientList[Series[Exp[Sum[(-1)^(k + 1) x^k/(k (1 - x^k) (1 - 2 x^k)), {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d (2^d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 32}]

G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k)*(1 - 2*x^k))).

a(n) ~ c * exp(2*sqrt(n) - 1/2) * 2^(n-1) / (A079555 * sqrt(Pi) * n^(3/4)), where c = exp(Sum_{k>=2} (-1)^(k-1)/(k*(2^(k-1)-1))) = 0.6602994483152065685... - Vaclav Kotesovec, Sep 15 2021

A383073 a(n) = [x^n] Product_{k=1..n} (1 + x^k)^binomial(n,k).

Original entry on oeis.org

1, 1, 2, 11, 69, 552, 5133, 53804, 626440, 7979043, 110074741, 1631532542, 25813521836, 433619035254, 7698641650937, 143908414079881, 2822753485000135, 57930283521990154, 1240695879627856673, 27666701629865989070, 641049490249340264699

Offset: 0

Vaclav Kotesovec, Apr 15 2025

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Cf. A360660.

Cf. A026007, A102866, A344130, A369578.

Table[SeriesCoefficient[Product[(1 + x^k)^Binomial[n, k], {k, 1, n}], {x, 0, n}], {n, 0, 20}]

A305652 Expansion of Product_{k>=1} (1 + x^k)^(2^(k-1)-1).

Original entry on oeis.org

1, 0, 1, 3, 7, 18, 41, 99, 227, 538, 1236, 2872, 6597, 15166, 34669, 79150, 180011, 408616, 925015, 2089607, 4709937, 10595275, 23788174, 53312366, 119271967, 266399612, 594077742, 1322815256, 2941225084, 6530659320, 14481362803, 32070677496, 70937233268, 156721128440

Offset: 0

Ilya Gutkovskiy, Jun 07 2018

nonn

Weigh transform of A000225, shifted right one place.

Convolution of the sequences A081362 and A098407.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms

Cf. A000225, A001906, A011782, A081362, A098407, A102866, A110045.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(2^(i-1)-1, j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..40);  # Alois P. Heinz, Jun 07 2018

nmax = 33; CoefficientList[Series[Product[(1 + x^k)^(2^(k-1)-1), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 33; CoefficientList[Series[Exp[Sum[(-1)^(k + 1) x^(2 k)/(k (1 - x^k) (1 - 2 x^k)), {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d (2^(d - 1) - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 33}]

G.f.: Product_{k>=1} (1 + x^k)^A000225(k-1).
G.f.: Product_{k>=1} (1 + x^k)^(A011782(k)-1).
G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^(2*k)/(k*(1 - x^k)*(1 - 2*x^k))).
a(n) ~ 2^n * exp(sqrt(2*n) - 5/4 + c) / (sqrt(2*Pi) * 2^(3/4) * n^(3/4)), where c = Sum_{k>=2} -(-1)^k / (k*(2^k-1)*(2^k-2)) = -0.07640757130267274170429705262846... - Vaclav Kotesovec, Jun 08 2018

A371482 Expansion of e.g.f. Product_{k>=1} (1 + x^k/k!)^(2^k).

Original entry on oeis.org

1, 2, 6, 32, 164, 1032, 7728, 59376, 522600, 4946768, 49680656, 540031296, 6195155744, 75183755584, 961596510272, 12909563309952, 181305865742240, 2657525771641664, 40594443765953472, 643987597483557888, 10601112599585001984, 180727870834447607808, 3185418524574895953152

Offset: 0

Ilya Gutkovskiy, Mar 25 2024

nonn

"EGJ" (unordered, element, labeled) transform of powers of 2.

C. G. Bower, Transforms (2)

Cf. A000079, A007837, A032315, A032316, A102866, A199163.

nmax = 22; CoefficientList[Series[Product[(1 + x^k/k!)^(2^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

cp's OEIS Frontend

A360634 Number T(n,k) of sets of nonempty words over binary alphabet with a total of n letters of which k are the first letter; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A261053 Expansion of Product_{k>=1} (1+x^k)^(k^k).

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A261052 Expansion of Product_{k>=1} (1+x^k)^(k!).

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A261051 Expansion of Product_{k>=1} (1+x^k)^(Lucas(k)).

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A216158 The total number of nonempty words in all length n finite languages on an alphabet of two letters.

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A319919 Expansion of Product_{k>=1} (1 + x^k)^(2^k-1).

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A383073 a(n) = [x^n] Product_{k=1..n} (1 + x^k)^binomial(n,k).

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A305652 Expansion of Product_{k>=1} (1 + x^k)^(2^(k-1)-1).

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