A246935 -id:A246935 - cp's OEIS frontend

A071109 Expansion of Product_{k>=1} 1/(1+2*x^k).

1, -2, 2, -6, 14, -26, 50, -102, 214, -426, 834, -1678, 3398, -6778, 13482, -27022, 54198, -108306, 216346, -432878, 866334, -1732386, 3463626, -6927926, 13858350, -27715378, 55426002, -110855030, 221719582, -443433610, 886848930, -1773709078, 3547455846

Offset: 0

Sharon Sela (sharonsela(AT)hotmail.com), May 27 2002

sign

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

Cf. A070933, A032302, A000700, A246935, A261562, A261566, A261582.

nmax = 40; CoefficientList[Series[Product[1/(1 + 2*x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 25 2015 *)
nmax = 40; CoefficientList[Series[Exp[Sum[(-1)^k*2^k/k*x^k/(1-x^k), {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 25 2015 *)
(O[x]^30 + 3/QPochhammer[-2, x])[[3]] (* Vladimir Reshetnikov, Nov 20 2015 *)

a(n) ~ c * (-2)^n, where c = Product_{j>=1} 1/(1-1/(-2)^j) = 1/QPochhammer[-1/2,-1/2] = 0.8259519860658427384636116224100201356301... . - Vaclav Kotesovec, Aug 25 2015

G.f.: Sum_{i>=0} (-2)^i*x^i/Product_{j=1..i} (1 - x^j). - Ilya Gutkovskiy, Apr 13 2018

More terms from Vaclav Kotesovec, Aug 25 2015

A255970 Number T(n,k) of partitions of n into parts of exactly k sorts; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 2, 2, 0, 3, 8, 6, 0, 5, 24, 42, 24, 0, 7, 60, 198, 264, 120, 0, 11, 144, 780, 1848, 1920, 720, 0, 15, 320, 2778, 10512, 18840, 15840, 5040, 0, 22, 702, 9342, 53184, 146760, 208080, 146160, 40320, 0, 30, 1486, 30186, 250128, 999720, 2129040, 2479680, 1491840, 362880

Offset: 0

Alois P. Heinz, Mar 12 2015

			T(3,1) = 3: 1a1a1a, 2a1a, 1a.
T(3,2) = 8: 1a1a1b, 1a1b1a, 1b1a1a, 1b1b1a, 1b1a1b, 1a1b1b, 2a1b, 2b1a.
T(3,3) = 6: 1a1b1c, 1a1c1b, 1b1a1c, 1b1c1a, 1c1a1b, 1c1b1a.
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  2,   2;
  0,  3,   8,    6;
  0,  5,  24,   42,    24;
  0,  7,  60,  198,   264,    120;
  0, 11, 144,  780,  1848,   1920,    720;
  0, 15, 320, 2778, 10512,  18840,  15840,   5040;
  0, 22, 702, 9342, 53184, 146760, 208080, 146160, 40320;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-1 give: A000007, A000041 (for n>0).
Main diagonal gives A000142.
Row sums give A278644.
Cf. A246935, A256130, A319600.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))
    end:
T:= (n, k)-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..10);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k] + If[i>n, 0, k*b[n-i, i, k]]]]; T[n_, k_] := Sum[b[n, n, k -i]*(-1)^i* Binomial[k, i], {i, 0, k}]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 22 2016, after Alois P. Heinz *)

T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A246935(n,k-i).

T(n,k) = k! * A256130(n,k).

A261561 Expansion of Product_{k>=1} (1/(1 - 2*x^k))^k.

Original entry on oeis.org

1, 2, 8, 22, 64, 162, 424, 1022, 2480, 5770, 13336, 30046, 67184, 147554, 321592, 692278, 1479568, 3133474, 6596008, 13788606, 28679264, 59335530, 122256456, 250875550, 513116864, 1046190786, 2127557592, 4316282006, 8739096992, 17661731138, 35639764536

Offset: 0

Vaclav Kotesovec, Aug 24 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..2000 (first 1001 terms from Vaclav Kotesovec)

Cf. A000219, A070933, A246935, A261562, A261563, A261565.

b:= proc(n, i) option remember;  `if`(n=0, 1, `if`(i<1, 0,
      add(2^j*binomial(i+j-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, Sep 21 2018

nmax = 50; CoefficientList[Series[Product[(1/(1 - 2*x^k))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 50; CoefficientList[Series[Exp[Sum[2^k/k*x^k/(1 - x^k)^2, {k, 1, nmax}]], {x, 0, nmax}], x]

{a(n) = polcoeff( exp( sum(m=1,n,x^m/m * sumdiv(m,d,2^d*m^2/d^2) ) +x*O(x^n)),n)}
for(n=0,40,print1(a(n),", ")) \\ Paul D. Hanna, Sep 30 2015

a(n) ~ c * 2^n, where c = Product_{j>=1} 1/(1 - 1/2^j)^(j+1) = 34.7387234654851595844514193757064296508992247003230539635669599773458896...

G.f.: exp( Sum_{n>=1} x^n/n * Sum_{d|n} 2^d * n^2/d^2 ). - Paul D. Hanna, Sep 30 2015

A261565 Expansion of Product_{k>=1} (1/(1 - 3*x^k))^k.

Original entry on oeis.org

1, 3, 15, 54, 201, 672, 2268, 7266, 23208, 72414, 224652, 688929, 2103975, 6386907, 19337091, 58367817, 175905741, 529331190, 1591515297, 4781575074, 14359673454, 43108645230, 129387584991, 388283978589, 1165099808574, 3495782937135, 10488322595625

Offset: 0

Vaclav Kotesovec, Aug 24 2015

nonn

In general, for z > 1 or z < -1, if g.f. = Product_{k>=1} (1/(1 - z*x^k))^k, then a(n) ~ c * z^n, where c = Product_{j>=1} 1/(1 - 1/z^j)^(j+1).

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A000219, A242587, A246935, A261561.

nmax = 40; CoefficientList[Series[Product[(1/(1 - 3*x^k))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[Exp[Sum[3^k/k*x^k/(1 - x^k)^2, {k, 1, nmax}]], {x, 0, nmax}], x]

{a(n) = polcoeff( exp( sum(m=1, n, x^m/m * sumdiv(m, d, 3^d * m^2/d^2) ) +x*O(x^n)), n)}
for(n=0, 40, print1(a(n), ", ")) \\ Paul D. Hanna, Sep 30 2015

a(n) ~ c * 3^n, where c = Product_{j>=1} 1/(1 - 1/3^j)^(j+1) = 4.1269357592430271005054028580646705856298720432004233223482475759761040273...

G.f.: exp( Sum_{n>=1} x^n/n * Sum_{d|n} 3^d * n^2/d^2 ). - Paul D. Hanna, Sep 30 2015

A246937 Number of partitions of n into 5 sorts of parts.

Original entry on oeis.org

1, 5, 30, 155, 805, 4055, 20455, 102455, 513230, 2567230, 12841130, 64211380, 321082905, 1605444405, 8027354055, 40136925680, 200685295955, 1003427268205, 5017139711105, 25085702537730, 125428529603755, 627142668099880, 3135713425289030, 15678567227192655

Offset: 0

Alois P. Heinz, Sep 08 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=5 of A246935.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1) +`if`(i>n, 0, 5*b(n-i, i))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);

(O[x]^20 - 4/QPochhammer[5, x])[[3]] (* Vladimir Reshetnikov, Nov 20 2015 *)
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, 5 b[n-i, i]]]];
a[n_] := b[n, n];
a /@ Range[0, 25] (* Jean-François Alcover, Jan 02 2021, after Alois P. Heinz *)

G.f.: Product_{i>=1} 1/(1-5*x^i).
a(n) ~ c * 5^n, where c = Product_{k>=1} 1/(1-1/5^k) = 1.3152135557353452193080... . - Vaclav Kotesovec, Mar 19 2015
G.f.: Sum_{i>=0} 5^i*x^i/Product_{j=1..i} (1 - x^j). - Ilya Gutkovskiy, Apr 12 2018

A265974 Expansion of Product_{k>=1} 1/(1 - 3kx^k).

Original entry on oeis.org

1, 3, 15, 54, 210, 699, 2484, 7995, 26610, 84186, 269940, 839238, 2634579, 8098194, 25032282, 76388265, 233791104, 709501596, 2157488730, 6523204836, 19747491810, 59558682132, 179762506329, 541222906812, 1630300772106, 4902697929306, 14748249476553

Offset: 0

Vaclav Kotesovec, Dec 19 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A006906, A265951, A265975, A265976.

Cf. A246935, A242587.

b:= proc(n, i) option remember; `if`(n=0 or i=1,
      3^n, b(n, i-1) +i*3*b(n-i, min(n-i, i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..32);  # Alois P. Heinz, Aug 23 2019

nmax=40; CoefficientList[Series[Product[1/(1-3*k*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c * 3^n, where c = Product_{m>=2} 1/(1 - m/3^(m-1)) = 5.86277744540963226378877460838259757442241952947887939654316926419876...

A265975 Expansion of Product_{k>=1} 1/(1 - 4kx^k).

Original entry on oeis.org

1, 4, 24, 108, 512, 2164, 9464, 39004, 163008, 663588, 2713752, 10954764, 44328512, 178160724, 716821752, 2874497660, 11532111232, 46187508676, 185028540696, 740595436652, 2964628293504, 11862432443764, 47467812675320, 189902835709212, 759756868215872

Offset: 0

Vaclav Kotesovec, Dec 19 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A006906, A265951, A265974, A265976.

Cf. A246935, A246936.

b:= proc(n, i) option remember; `if`(n=0 or i=1,
      4^n, b(n, i-1) +i*4*b(n-i, min(n-i, i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..32);  # Alois P. Heinz, Aug 23 2019

nmax=40; CoefficientList[Series[Product[1/(1-4*k*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c * 4^n, where c = Product_{m>=2} 1/(1 - m/4^(m-1)) = 2.700170514502619666262858845683166558216386190684736249639219328278569...

A265976 Expansion of Product_{k>=1} 1/(1 - 5kx^k).

Original entry on oeis.org

1, 5, 35, 190, 1070, 5525, 29080, 147485, 752790, 3789170, 19105800, 95794930, 480650335, 2406018490, 12047084370, 60264282575, 301493182380, 1507758356660, 7540528037090, 37705593514220, 188545393000350, 942756783659980, 4713958620697385, 23570092258449540

Offset: 0

Vaclav Kotesovec, Dec 19 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A006906, A265951, A265974, A265975.

Cf. A246935, A246937.

b:= proc(n, i) option remember; `if`(n=0 or i=1,
      5^n, b(n, i-1) +i*5*b(n-i, min(n-i, i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..32);  # Alois P. Heinz, Aug 23 2019

nmax=40; CoefficientList[Series[Product[1/(1-5*k*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c * 5^n, where c = Product_{m>=2} 1/(1 - m/5^(m-1)) = 1.977268427518901757865749340705853730491796767544158844539130847296...

A246938 Number of partitions of n into 6 sorts of parts.

Original entry on oeis.org

1, 6, 42, 258, 1590, 9582, 57786, 347010, 2083902, 12505470, 75044202, 450278106, 2701739022, 16210513806, 97263509010, 583581545466, 3501491846046, 21008954050422, 126053739826530, 756322456907130, 4537934834757702, 27227609116759302, 163365655261094322

Offset: 0

Alois P. Heinz, Sep 08 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=6 of A246935.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1) +`if`(i>n, 0, 6*b(n-i, i))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);

(O[x]^20 - 5/QPochhammer[6, x])[[3]] (* Vladimir Reshetnikov, Nov 20 2015 *)

G.f.: Product_{i>=1} 1/(1-6*x^i).
a(n) ~ c * 6^n, where c = Product_{k>=1} 1/(1-1/6^k) = 1.2411756627857248707756... . - Vaclav Kotesovec, Mar 19 2015
G.f.: Sum_{i>=0} 6^i*x^i/Product_{j=1..i} (1 - x^j). - Ilya Gutkovskiy, Apr 12 2018

A246939 Number of partitions of n into 7 sorts of parts.

Original entry on oeis.org

1, 7, 56, 399, 2849, 19999, 140441, 983535, 6887986, 48219486, 337559586, 2362943030, 16540767131, 115785555389, 810500055939, 5673501716540, 39714520225149, 278001650902563, 1946011613977669, 13622081363362570, 95354569947550935, 667481990092883448

Offset: 0

Alois P. Heinz, Sep 08 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=7 of A246935.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1) +`if`(i>n, 0, 7*b(n-i, i))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);

(O[x]^20 - 6/QPochhammer[7, x])[[3]] (* Vladimir Reshetnikov, Nov 20 2015 *)

G.f.: Product_{i>=1} 1/(1-7*x^i).
a(n) ~ c * 7^n, where c = Product_{k>=1} 1/(1-1/7^k) = 1.1950352398308474540223... . - Vaclav Kotesovec, Mar 19 2015
G.f.: Sum_{i>=0} 7^i*x^i/Product_{j=1..i} (1 - x^j). - Ilya Gutkovskiy, Apr 12 2018

cp's OEIS Frontend

A071109 Expansion of Product_{k>=1} 1/(1+2*x^k).

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A255970 Number T(n,k) of partitions of n into parts of exactly k sorts; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A261561 Expansion of Product_{k>=1} (1/(1 - 2*x^k))^k.

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A261565 Expansion of Product_{k>=1} (1/(1 - 3*x^k))^k.

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A246937 Number of partitions of n into 5 sorts of parts.

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A265974 Expansion of Product_{k>=1} 1/(1 - 3*k*x^k).

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A265975 Expansion of Product_{k>=1} 1/(1 - 4*k*x^k).

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A265976 Expansion of Product_{k>=1} 1/(1 - 5*k*x^k).

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A265974 Expansion of Product_{k>=1} 1/(1 - 3kx^k).

A265975 Expansion of Product_{k>=1} 1/(1 - 4kx^k).

A265976 Expansion of Product_{k>=1} 1/(1 - 5kx^k).