A074141 -id:A074141 - cp's OEIS frontend

A077802 Sum of products of parts increased by 1 in hook partitions of n, where hook partitions are of the form h*1^(n-h).

1, 2, 7, 18, 41, 88, 183, 374, 757, 1524, 3059, 6130, 12273, 24560, 49135, 98286, 196589, 393196, 786411, 1572842, 3145705, 6291432, 12582887, 25165798, 50331621, 100663268, 201326563, 402653154, 805306337, 1610612704

Offset: 0

Alford Arnold, Dec 02 2002

It is not clear whether a(0) should be 1 or 0; this depends on whether the empty partition is a hook partition. By strict interpretation of the definition above, it is not; and except for n=0, there are exactly n hook partitions for each n. On the other hand, if defined as "a partition in whose Ferrers diagram every point is on the first row or column", the empty partition is a hook partition. - Franklin T. Adams-Watters, Jul 11 2009

			The hook partitions of 4 are 4, 3+1, 2+1+1, 1+1+1+1; the corresponding products when parts are increased by 1 are 5, 8, 12, 16; and their sum is a(4) = 41.

Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A074141, A055010 (first differences), A042950 (second differences).
Cf. A132048.
Same as A095151 except for a(0). - Franklin T. Adams-Watters, Jul 11 2009

s=0;lst={1};Do[s+=(s-n);AppendTo[lst, Abs[s]], {n, 2, 4!}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 10 2008 *)

a(n)=if(n>0, 3*2^n - n - 3, 1) \\ Charles R Greathouse IV, Aug 08 2016

From Vladeta Jovovic, Dec 05 2002: (Start)
a(n) = 3*2^n - n - 3, n > 0.
G.f.: x*(2-x)/(1-2*x)/(1-x)^2.
Recurrence: a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3). (End)
Row sums of triangle A132048. Equals binomial transform of [1, 1, 4, 2, 4, 2, 4, 2, 4, ...]. - Gary W. Adamson, Aug 08 2007
a(n) = A125128(n) + A000225(n), n >= 1. - Miquel Cerda, Aug 07 2016

More terms from John W. Layman, Dec 05 2002

A265836 Expansion of Product_{k>=1} 1/(1 - k(k+1)x^k).

Original entry on oeis.org

1, 2, 10, 32, 120, 342, 1206, 3320, 10604, 29578, 88342, 239400, 702020, 1863654, 5262650, 13948824, 38427192, 100244162, 272822282, 703972024, 1883948848, 4839944150, 12779850278, 32548367784, 85335644100, 215826029018, 560407835934, 1412632075328

Offset: 0

Vaclav Kotesovec, Dec 16 2015

nonn

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

Cf. A074141, A077335, A092485, A305204.

b:= proc(n, i) option remember; `if`(n=0 or i=1,
      2^n, b(n, i-1)+(1+i)*i*b(n-i, min(n-i, i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..33);  # Alois P. Heinz, Aug 16 2019

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

a(n) ~ c * 6^(n/2), where
c = 79.0418032646837469192452349...... if n is even,
c = 78.4480460169710091436913691...... if n is odd.

A267008 Expansion of Product_{k>=1} (1 + (k+1)*x^k).

Original entry on oeis.org

1, 2, 3, 10, 13, 28, 58, 90, 146, 260, 481, 688, 1168, 1748, 2863, 4726, 6938, 10412, 16140, 23746, 35702, 55812, 79032, 116758, 168779, 247006, 350310, 513410, 744286, 1045466, 1485685, 2098780, 2935416, 4137878, 5746618, 8027612, 11343706, 15487222, 21418682

Offset: 0

Vaclav Kotesovec, Jan 08 2016

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A022629, A074141, A267007.

b:= proc(n, i) option remember; `if`(i*(i+1)/2 b(n$2):
seq(a(n), n=0..42);  # Alois P. Heinz, Aug 15 2019

nmax = 50; CoefficientList[Series[Product[1+(k+1)*x^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 50; poly = ConstantArray[0, nmax+1]; poly[[1]] = 1; poly[[2]] = 2; Do[Do[poly[[j+1]] += (k+1)*poly[[j-k+1]], {j, nmax, k, -1}];, {k, 2, nmax}]; poly

A079308 For a partition P of a positive integer, let f(P) be the product of k+1, over all parts k in P. Let a(n,r) be the sum of f(P) over all partitions P of n with smallest part r. Sequence gives table of a(n,r) for 1 <= r <= n, in the order a(1,1); a(2,1), a(2,2); a(3,1), a(3,2), a(3,3); ...

Original entry on oeis.org

2, 4, 3, 14, 0, 4, 36, 9, 0, 5, 100, 12, 0, 0, 6, 236, 42, 16, 0, 0, 7, 602, 54, 20, 0, 0, 0, 8, 1368, 195, 24, 25, 0, 0, 0, 9, 3242, 246, 92, 30, 0, 0, 0, 0, 10, 7240, 759, 112, 35, 36, 0, 0, 0, 0, 11, 16386, 1134, 232, 40, 42, 0, 0, 0, 0, 0, 12, 35692, 2859, 528, 170, 48, 49, 0, 0, 0, 0, 0, 13

Offset: 1

Alford Arnold, Feb 09 2003

			The partitions with minimal part 3 begin 3, 3+3, 4+3, 5+3, 6+3, 3+3+3, ... which yield the following values of f: 4, 16, 20, 24, 28, 64, ... therefore the 3rd column of our table begins 4,0,0,16,20,24,(28+64)=92,...
Triangle a(n,r) begins:
:    2;
:    4,   3;
:   14,   0,   4;
:   36,   9,   0,  5;
:  100,  12,   0,  0,  6;
:  236,  42,  16,  0,  0, 7;
:  602,  54,  20,  0,  0, 0, 8;
: 1368, 195,  24, 25,  0, 0, 0, 9;
: 3242, 246,  92, 30,  0, 0, 0, 0, 10;
: 7240, 759, 112, 35, 36, 0, 0, 0,  0, 11;

Alois P. Heinz, Rows n = 1..141, flattened

Cf. A074139, A074141 (row sums).

b:= proc(n, k) option remember; `if`(n=0, 1,
      `if`(k>n, 0, b(n, k+1) +(k+1)*b(n-k, k)))
    end:
a:= (n, k)-> b(n,k)-b(n, k+1):
seq(seq(a(n, k), k=1..n), n=1..12);  # Alois P. Heinz, May 22 2015

a[n_, r_] := Which[r>n, 0, r==n, n+1, True, a[n, r]=(r+1)Sum[a[n-r, s], {s, r, n-r}]]; Flatten[Table[a[n, r], {n, 1, 12}, {r, 1, n}]]

Edited by Dean Hickerson, Feb 11 2003

Offset changed to 1 by Alois P. Heinz, May 22 2015

A078436 Triangle read by rows in which n-th row counts multisets associated with hook partitions.

Original entry on oeis.org

1, 2, 0, 3, 4, 0, 4, 6, 8, 0, 5, 8, 12, 16, 0, 6, 10, 16, 24, 32, 0, 7, 12, 20, 32, 48, 64, 0, 8, 14, 24, 40, 64, 96, 128, 0, 9, 16, 28, 48, 80, 128, 192, 256, 0, 10, 18, 32, 56, 96, 160, 256, 384, 512, 0, 11, 20, 36, 64, 112, 192, 320, 512, 768, 1024, 0, 12, 22, 40, 72, 128

Offset: 1

Alford Arnold, Dec 30 2002

Row sums appear to be A077802. When more general partition types are included, such as 22^(n-4) yielding 9 18 36 72 ..., the array row sums becomes 1,2,7,18,50,118,301,... in agreement with A074141.

			Triangle begins 1; 2,0; 3,4,0; 4,6,8,0; 5,8,12,16,0; ...
a(13) = 12 because we find 1 + 3 + 4 + 3 + 1 multisets of type 21^(n-2): they are 4; 14,24,34; 114,124,134,234; 1124,1134,1234; and 11234

Cf. A077802, A074139, A074141.

G.f.: x*y*(2-x)/(1-2*x*y)/(1-x)^2. - Vladeta Jovovic, Dec 31 2002

More terms from Vladeta Jovovic, Dec 31 2002

A079274 Number of divisors associated with the cyclic cases within the n-th group of least prime signatures.

Original entry on oeis.org

1, 0, 3, 4, 14, 18, 65, 82, 253, 378, 953, 1460, 3667, 5480, 12917, 20582, 45130, 71970, 156510, 248020, 524446, 846962, 1733252, 2806484, 5702101, 9186010, 18375867, 29872508, 58760991, 95500008, 186871472, 302966474, 587603027, 956192230

Offset: 0

Alford Arnold, Feb 06 2003

			The number of divisors doubles each time we create a new "matching" partition by appending "1" to a given partition. Therefore we may write
..1..0..3..4..14..18..65..82..253..378..953...
+ ...2..4.14..36.100.236.602.1368.3242.7240...
= 1..2..7.18..50.118.301.684.1621.3620.8193...

Cf. A074141.

a(n) = A074141(n) - 2* A074141(n-1) for n > 0 and a(0) = 1

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 27 2004

A319110 Expansion of Product_{k>=1} 1/(1 - (k - 1)*x^k).

Original entry on oeis.org

1, 0, 1, 2, 4, 6, 13, 18, 37, 56, 101, 152, 285, 410, 713, 1118, 1830, 2780, 4618, 6934, 11278, 17092, 26894, 40822, 64435, 96372, 149299, 225104, 345131, 515394, 788176, 1169962, 1772957, 2632458, 3950365, 5849260, 8748993, 12867848, 19135894, 28126614, 41598695

Offset: 0

Ilya Gutkovskiy, Sep 10 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..8190
Vaclav Kotesovec, Closed form of the asymptotics

Cf. A006906, A034008, A052847, A074141, A267007.

b:= proc(n, i) option remember; `if`(n=0 or i=1,
       0^n, b(n, i-1)+(i-1)*b(n-i, min(n-i, i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..42);  # Alois P. Heinz, Aug 19 2019

nmax = 40; CoefficientList[Series[Product[1/(1 - (k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[Exp[Sum[Sum[(j - 1)^k x^(j k)/k, {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (d - 1)^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 40}]

G.f.: exp(Sum_{k>=1} Sum_{j>=1} (j - 1)^k*x^(j*k)/k).
From Vaclav Kotesovec, Sep 11 2018: (Start)
a(n) ~ c * 2^(2*n/5), where
c = 28108804.248904780960402246466460350520790117596512766842168... if mod(n,5) = 0
c = 28108804.010850549080284030388905319123062152339902207992657... if mod(n,5) = 1
c = 28108804.067769166625741650205643600577757560110636366636106... if mod(n,5) = 2
c = 28108804.083581827971851596540314974909801290757084687583764... if mod(n,5) = 3
c = 28108804.058853893104368046896759214442695016905368229405793... if mod(n,5) = 4
(End)

A368090 Triangle read by rows. T(n, k) = Sum_{p in P(n, k)} Product_{r in p}(r + 1), where P(n, k) are the partitions of n with length k.

Original entry on oeis.org

1, 0, 2, 0, 3, 4, 0, 4, 6, 8, 0, 5, 17, 12, 16, 0, 6, 22, 34, 24, 32, 0, 7, 43, 71, 68, 48, 64, 0, 8, 52, 122, 142, 136, 96, 128, 0, 9, 86, 197, 325, 284, 272, 192, 256, 0, 10, 100, 350, 502, 650, 568, 544, 384, 512

Offset: 0

Peter Luschny, Dec 11 2023

			Triangle T(n, k) starts:
  [0] [1]
  [1] [0,  2]
  [2] [0,  3,   4]
  [3] [0,  4,   6,   8]
  [4] [0,  5,  17,  12,  16]
  [5] [0,  6,  22,  34,  24,  32]
  [6] [0,  7,  43,  71,  68,  48,  64]
  [7] [0,  8,  52, 122, 142, 136,  96, 128]
  [8] [0,  9,  86, 197, 325, 284, 272, 192, 256]
  [9] [0, 10, 100, 350, 502, 650, 568, 544, 384, 512]

Cf. A238963, A368091, A074141 (row sums).

def T(n, k):
    return sum(product(r+1 for r in p) for p in Partitions(n, length=k))
for n in range(10): print([T(n, k) for k in range(n + 1)])

A368091 Triangle read by rows. T(n, k) = Sum_{p in P(n, k)} Product_{r in p} r, where P(n, k) are the partitions of n with length k.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 3, 2, 1, 0, 4, 7, 2, 1, 0, 5, 10, 7, 2, 1, 0, 6, 22, 18, 7, 2, 1, 0, 7, 28, 34, 18, 7, 2, 1, 0, 8, 50, 62, 50, 18, 7, 2, 1, 0, 9, 60, 121, 86, 50, 18, 7, 2, 1, 0, 10, 95, 182, 189, 118, 50, 18, 7, 2, 1

Offset: 0

Peter Luschny, Dec 11 2023

			Table T(n, k) starts:
  [0] [1]
  [1] [0, 1]
  [2] [0, 2,  1]
  [3] [0, 3,  2,   1]
  [4] [0, 4,  7,   2,  1]
  [5] [0, 5, 10,   7,  2,  1]
  [6] [0, 6, 22,  18,  7,  2,  1]
  [7] [0, 7, 28,  34, 18,  7,  2, 1]
  [8] [0, 8, 50,  62, 50, 18,  7, 2, 1]
  [9] [0, 9, 60, 121, 86, 50, 18, 7, 2, 1]

Cf. A368090, A074141, A023855, A006906 (row sums).

def T(n, k):
    return sum(product(r for r in p) for p in Partitions(n, length=k))
for n in range(10): print([T(n, k) for k in range(n + 1)])

cp's OEIS Frontend

A077802 Sum of products of parts increased by 1 in hook partitions of n, where hook partitions are of the form h*1^(n-h).

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A265836 Expansion of Product_{k>=1} 1/(1 - k*(k+1)*x^k).

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Maple

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A267008 Expansion of Product_{k>=1} (1 + (k+1)*x^k).

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Maple

Mathematica

A079308 For a partition P of a positive integer, let f(P) be the product of k+1, over all parts k in P. Let a(n,r) be the sum of f(P) over all partitions P of n with smallest part r. Sequence gives table of a(n,r) for 1 <= r <= n, in the order a(1,1); a(2,1), a(2,2); a(3,1), a(3,2), a(3,3); ...

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Maple

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A078436 Triangle read by rows in which n-th row counts multisets associated with hook partitions.

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Formula

Extensions

A079274 Number of divisors associated with the cyclic cases within the n-th group of least prime signatures.

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Extensions

A319110 Expansion of Product_{k>=1} 1/(1 - (k - 1)*x^k).

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Maple

Mathematica

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A368090 Triangle read by rows. T(n, k) = Sum_{p in P(n, k)} Product_{r in p}(r + 1), where P(n, k) are the partitions of n with length k.

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SageMath

A368091 Triangle read by rows. T(n, k) = Sum_{p in P(n, k)} Product_{r in p} r, where P(n, k) are the partitions of n with length k.

A265836 Expansion of Product_{k>=1} 1/(1 - k(k+1)x^k).