William J. Keith - cp's OEIS frontend

A366920 a(n) is the number times a Dyck path in an m X m box of any size has area n, counted to the lower right.

1, 1, 0, 1, 1, 0, 1, 2, 1, 1, 1, 3, 3, 3, 2, 2, 5, 6, 7, 7, 5, 6, 8, 12, 15, 18, 16, 16, 15, 17, 24, 32, 40, 43, 45, 45, 42, 44, 53, 69, 87, 104, 115, 126, 125, 124, 124, 136, 160, 198, 240, 282, 321, 345, 360, 365, 367, 382, 417, 482, 574, 682, 791, 895, 976

Offset: 0

William J. Keith, Oct 28 2023

nonn

A Dyck path in an m X m grid is a set of up steps U and right steps R from the lower left corner to the upper right corner, staying weakly above the diagonal.
For this statistic, count the boxes below and to the right of the path.
The first time an area appears in two different squares is at size 15, which appears in the 4 X 4 box below UUURURRR and in the 5 X 5 box below URURURURUR.

			The 0 X 0 box yields the trivial (empty) path of area 0.
The 1 X 1 box yields one Dyck path of area 1 (UR).
The 2 X 2 box yields one Dyck path each of area 3 (URUR) and 4 (UURR).
The 3 X 3 box yields one Dyck path of area 6 (URURUR), two of area 7 (UURRUR and URUURR), and one each of area 8 (UURURR) and 9 (UUURRR).

Cf. A143951, A227543.

G.f.: 1 + q + q^3 + q^4 + q^6 + 2q^7 + ...

To construct the g.f., take A(x,q) as defined in A227543, and replace each instance of x^k with q^(k*(k+1)/2).

a(45)-a(65) from Alois P. Heinz, Oct 29 2023

A298411 Coefficients of q^(-1/24)*eta(4q)^(1/2).

Original entry on oeis.org

1, -2, -10, -20, -90, 132, -836, 6040, 2310, 60180, 180308, 1662568, -2995620, 24401320, 44072120, -102437328, 19390406, 2649221300, -10584460060, 14475802440, -228570333836, -815899620616, 2088529753800, -5590702681520, -100828534100580, -172013432412024

Offset: 0

William J. Keith, Jan 18 2018

The q^(kn) term of any single factor of the product (1-(4q)^k)^(1/2) is (-2)*A000108(n-1). Hence these numbers are related to the Catalan numbers A000108 by a partition-based convolution.
Sequence appears to be positive and negative roughly half the time.
This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1/2, g(n) = 4^n. - Seiichi Manyama, Apr 20 2018

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A000108, A271235, A298994.

Expansion of Product_{n>=1} (1 - ((b^2)*x)^n)^(1/b): A010815 (b=1), this sequence (b=2), A303152 (b=3), A303153 (b=4), A303154 (b=5).

Series[Product[(1 - (4 q)^k)^(1/2), {k, 1, 100}], {q, 0, 100}]

q='q+O('q^99); Vec(eta(4*q)^(1/2)) \\ Altug Alkan, Apr 20 2018

G.f.: Product_{k>=1} (1 - (4x)^k)^(1/2).

A253926 a(n) is the excess of the number of Collatz permutations of length n (with first index 15) over the n-th Fibonacci number.

Original entry on oeis.org

1, 2, 3, 3, 4, 6, 7, 9, 12, 15, 19, 24, 30, 39, 49, 61, 77, 96

Offset: 15

William J. Keith, Jan 19 2015

A permutation is Collatz if for some n it is the sequence of ranks of terms prior to a power of 2 generated by the Collatz function C(n) = n/2 if n even, (3n+1) if n odd. For instance, iteration of the Collatz function on 12 generates 12, 6, 3, 10, 5, which is then followed by 16, so (5,3,1,4,2) is Collatz. Among n = 1 to 14, the number of Collatz permutations is the n-th Fibonacci number; thereafter, there is an increasing excess. This sequence counts the excess.

Michael Albert, Bjarki Gudmundsson and Henning Ulfarsson, Collatz meets Fibonacci. arXiv:1404.3054 [math.CO], 2014-2015. Table on page 4.

A000045, the Fibonacci numbers, gives the number of Collatz permutations for n <= 14.

A242168 Decimal expansion of the integral of the q-Pochhammer symbol (reciprocal of the partition function) over the real interval -1 to 1.

Original entry on oeis.org

1, 2, 8, 8, 3, 0, 0, 8, 8, 8, 6, 7, 3, 9, 2, 1, 2, 3, 0, 1, 8, 0, 9, 0, 1, 4, 9, 3, 9, 3, 0, 9, 6, 3, 4, 4, 4, 2, 2, 5, 8, 7, 3, 8, 0, 7, 1, 3, 8, 7, 9, 6, 1, 9, 5, 0, 3, 2, 0, 1, 4, 9, 4, 2, 6, 9, 8, 6, 4, 4, 2, 4, 1, 8, 5, 2, 0, 4, 9, 7, 8, 8, 7, 6, 8, 2, 0, 9, 3, 4, 4, 4, 4, 1, 1, 1, 3, 3, 9, 8, 1, 3, 6, 3, 3

Offset: 1

William J. Keith, May 05 2014

As a function, the q-Pochhammer symbol is an irregularly left-skewed bell curve. It has limiting value 0 at -1 and 1, and its maximum is at -0.411248... (decimal value given by A143441).

			1.2883008886739212301809014939309634442258738...

Vaclav Kotesovec, The integration of q-series
Vaclav Kotesovec, Graph of the area below a curve

Cf. A010815, A143441, A258232.

evalf(4*sqrt(3/23)*Pi * (2*sinh(sqrt(23)*Pi/6) + sqrt(2)*sinh(sqrt(23)*Pi/4)) / (2*cosh(sqrt(23)*Pi/3)-1), 120); # Vaclav Kotesovec, Jun 02 2015

NIntegrate[QPochhammer[q, q], {q, -1, 1}, WorkingPrecision -> 45]
RealDigits[4*Sqrt[3/23]*Pi*(2*Sinh[Sqrt[23]*Pi/6] + Sqrt[2]*Sinh[Sqrt[23]*Pi/4]) / (2*Cosh[Sqrt[23]*Pi/3]-1), 10, 105][[1]] (* Vaclav Kotesovec, Jun 02 2015 *)

eta2(q)=if(q==0,1,my(p=log(10^-38)/log(abs(q)),N=floor(sqrt(2*p/3)));sum(n=-N,N,(-1)^n*q^((3*n^2-n)/2),0.))
intnum(q=-.99999,.99999,eta2(q)) \\ Bill Allombert, May 06 2014

Equals 4*sqrt(3/23)*Pi * (2*sinh(sqrt(23)*Pi/6) + sqrt(2)*sinh(sqrt(23)*Pi/4)) / (2*cosh(sqrt(23)*Pi/3)-1). - Vaclav Kotesovec, Jun 02 2015

More digits from Vaclav Kotesovec, Jun 02 2015

A239445 Values n at which ratios of successive partition numbers approach 1 closer than the reciprocal of a whole number.

Original entry on oeis.org

2, 3, 11, 25, 39, 57, 78, 102, 130, 161, 195, 232, 273, 317, 365, 415, 469, 526, 587, 651, 718, 788, 862, 939, 1019, 1103, 1189, 1280, 1373, 1470, 1570, 1673, 1779, 1889, 2002, 2119, 2239, 2362, 2488, 2618, 2750, 2887, 3026, 3169, 3315, 3464, 3617, 3773, 3932, 4094, 4260, 4429, 4602, 4777, 4956

Offset: 1

William J. Keith, Mar 18 2014

The ratios of successive partition numbers p(n) / p(n-1) approach 1 monotonically, for n>1. a(k) gives the n for which p(n)/p(n+1) first equals or is less than 1+1/k.

			p(2)=2 and p(1)=1, so a(1) = 2, since p(2)/p(1) = 1+1/1.
p(3)=3 and p(2)=2, so a(2)=3, since p(3)/p(2) = 1+1/2.
p(11)=56 and p(10) = 42, so a(3) = 11, since p(11)/p(10) = 1+1/3.

Cf. A000041 (Partition numbers), A013661 (Pi^2 / 6).

AddDenom = 2;
Breaks = {};
For[n = 2, n < 10000, n++,
If[PartitionsP[n]/PartitionsP[n - 1] <= (1 + (1/AddDenom)),
  AppendTo[Breaks, n]; ADH = AddDenom + 1; AddDenom = ADH]
]
Breaks

Empirical quadratic fit to first 78 terms: ak^2 + bk + c, a ~ 1.64466, b ~ -0.3287, c ~ -0.66.

Leading term appears to approach 1.644... k^2, where the constant is zeta(2), Pi^2/6. This can probably be rigorously derived from the asymptotic expansion of the partition function, p(n) ~ 1/(4 n sqrt(3)) exp( Pi sqrt(2n/3)).

A238752 Number of nonisomorphic partial 1-differential posets up to rank n.

Original entry on oeis.org

1, 1, 1, 1, 2, 5, 35, 643, 44606, 29199636

Offset: 1

William J. Keith, Mar 04 2014

			For n<=4, label nodes with the partitions of n for convenience.
At n=5, the two possible posets are the Young poset (nodes and covering relations are the partitions of 5) and the poset constructed by covering the partitions (31), (22) and (211) with a common element, then giving each of those partitions another cover, and leaving all other nodes the same.

P. Byrnes, Structural aspects of differential posets, Ph.D. thesis, University of Minnesota, (2012)
Richard P. Stanley, Fabrizio Zanello, On the rank function of a differential poset, arXiv:1111.4371 [math.CO], 2011-2012.

A236680 Dimension of the space of spinors in n-dimensional real space.

Original entry on oeis.org

1, 2, 4, 4, 4, 4, 8, 8, 16, 32, 64, 64, 64, 64, 128, 128, 256, 512, 1024, 1024, 1024, 1024, 2048, 2048, 4096, 8192, 16384, 16384, 16384, 16384, 32768, 32768, 65536, 131072, 262144, 262144, 262144, 262144, 524288, 524288, 1048576, 2097152, 4194304

Offset: 1

Charles R Greathouse IV and William J. Keith, Jan 29 2014

a(n) = n only for n = 1, 2, 4, 8. These correspond to the four normed division algebras: the real numbers, the complex numbers, the quaternions, and the octonions.

All terms are powers of 2: a(n) = 2^A236916(n-1).

John Baez, John Baez on the number 8, 2008 Rankin Lecture (see frame at 38 minutes and 5 seconds).
Index entries for linear recurrences with constant coefficients, signature (2,-2,0,4,-8,8).

Cf. A236916.

Closely related to A034583 and A034584.

LinearRecurrence[{2,-2,0,4,-8,8},{1,2,4,4,4,4},50] (* Harvey P. Dale, May 05 2019 *)

Vec(x*(1+2*x^2+4*x^5)/((1-2*x^2)*(1+2*x^2)*(1-2*x+2*x^2)) + O(x^100)) \\ Colin Barker, Jan 30 2014

a(n) = 16*a(n-8) = 2*a(n-1) - 2*a(n-2) + 4*a(n-4) - 8*a(n-5) + 8*a(n-6).

G.f.: x*(1+2*x^2+4*x^5)/((1-2*x^2)*(1+2*x^2)*(1-2*x+2*x^2)). - Colin Barker, Jan 30 2014

A234937 Triangle read by rows of coefficients of polynomials generated by the Han/Nekrasov-Okounkov formula.

Original entry on oeis.org

1, 1, -1, 4, -5, 1, 18, -29, 12, -1, 120, -218, 119, -22, 1, 840, -1814, 1285, -345, 35, -1, 7920, -18144, 14674, -5205, 805, -51, 1, 75600, -196356, 185080, -79219, 16450, -1624, 70, -1, 887040, -2427312, 2515036, -1258628, 324569, -43568, 2954, -92, 1

Offset: 0

William J. Keith, Jan 01 2014

Coefficients of the polynomials p_n(b) defined by Product_{k>0} (1-q^k)^(b-1) = Sum n! p_n(b) q^n.
Each row is length 1+n, starting from n=0, and consists of the coefficients of one of the p_n(b).
A210590 is an unsigned version using the form preferred by Nekrasov and Okounkov. This is the form for which Guo-Niu Han's reference below gives the hooklength formula:
p_n(b) = Sum_{lambda partitioning n} Product_{h_{ij} in lambda} (1-b/(h_{ij}^2)).
Coefficients reduced mod 5 are those of 2 times Pascal's triangle and an alternating sign. Other primes have slightly more complex reduction behavior. See second link.
Lehmer's conjecture on the tau function states that the evaluation at b=25 (A000594) is never 0.
The general diagonal and column are probably of combinatorial interest.

			The coefficient of q^3 in the indeterminate power is (1/6) (18-29b+12b^2-b^3).

Seiichi Manyama, Rows n = 0..100, flattened
G.-N. Han, An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths, arXiv:0804.1849 [math.CO], 2008.
W. J. Keith, Polynomial analogues of Ramanujan congruences for Han's hooklength formula, arXiv:1109.1236 [math.CO], 2011-2012; Acta Arith. 160 (2013), 303-315.

Row entries sum to 0.
A210590 is the unsigned version.
Starting from row 0: final entry of row n, (-1)^n (A033999).
From row 1: next-to-last entry of row n, (-1)^(n-1) * n(3n-1)/2 (signed version of A000326).
First entry of row n, n! * p(n) (A053529).
Second entry of row n, -1 * n! * (sum of reciprocals of all parts in partitions of n) (negatives of A057623).
(Sum of absolute values of row entries)/n!: A000712.
Evaluations at various powers of b, divided by n!, enumerate multipartitions or powers of the eta function. Some special cases that appear in the OEIS:
b=0: A000041, the partition numbers,
b=2: A010815, from Euler's Pentagonal Number Theorem,
b=-1: A000712, partitions into 2 colors,
b=-11: A005758, reciprocal of the square root of the tau function,
b=-23: A006922, reciprocal of the tau function,
b=13: A000735, square root of the tau function,
b=25: A000594, Ramanujan's tau function.

nn=10;
Clear[b]; PolyTable = Table[0, {n, 1, nn}];
PolyTable[[1]]=1-b;
For[n = 2, n <= nn, n++,
PolyTable[[n]] = Simplify[(((n - 1)!)*(b - 1))*(Sum[
       PolyTable[[n - m]]*(-1*DivisorSigma[1, m]/((n - m)!)), {m, 1,
        n - 1}] + (-1*DivisorSigma[1, n]))]];
LongTable = Table[Table[
   Which[k == 0, PartitionsP[n]*n!, k > 0,
    Coefficient[Expand[PolyTable[[n]]], b^k]], {k, 0, n}], {n, 1, nn}];
Flatten[PrependTo[LongTable,1]]

E.g.f.: Product_{k>0} (1-q^k)^(b-1).

Recurrence: With p_0(b) = 1, p_n(b) = (n-1)!*(b-1)*Sum_{m=1..n} -sigma(m)*p_{n-m}(b) / (n-m)!, sigma being the divisor function.

A228117 Number of partitions of n that have hookset {1,2,...,k} for some k.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 4, 6, 7, 9, 10, 16, 14, 23, 24, 33, 33, 50, 50, 71, 75, 101, 103, 146, 151, 201, 211, 280, 292, 389, 409, 519, 573, 707, 765, 960, 1043, 1276, 1393, 1704, 1870, 2258, 2483, 2970, 3281, 3920, 4290, 5101, 5659, 6640, 7318, 8628, 9506, 11081

Offset: 0

William J. Keith, Aug 10 2013

nonn

It appears to be the case that the difference between entry a(2n-1) and a(2n) is substantially less than the difference between a(2n) and a(2n+1), after a few initial exceptions.

			a(7) = 6, counting the partitions (7), (43), (331), (322), (2221), and (111111).  The hooklengths of (7) are {1,2,3,4,5,6,7}, and the hooklengths of (322) are {1,1,2,2,3,4,5}.

Cf. A158291, the number of partitions which have hookset {1,2,...,n}, not counting multiplicities.

h:= proc(l) local n, s; n:=nops(l); s:= {seq(seq(1+l[i]-j
       +add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)};
       `if`(s={$1..max(s[], 0)}, 1, 0)
    end:
g:= (n, i, l)-> `if`(n=0 or i=1, h([l[], 1$n]), `if`(i<1, 0,
             g(n, i-1, l)+`if`(i>n, 0, g(n-i, i, [l[], i])))):
a:= n-> g(n$2, []):
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 12 2013

<< "Combinatorica`"
HookSet[Lambda_] := Module[{i, j, k, HookHolder},
  HookHolder = {};
  HS = {};
  For[i = 1, i < Length[Lambda] + 1, i++,
   For[j = 1, j < Lambda[[i]] + 1, j++,
    CurrentHook =
     Lambda[[i]] - j + TransposePartition[Lambda][[j]] - i + 1;
    If[! MemberQ[HS, CurrentHook],
     HookHolder = Append[HS, CurrentHook]; HS = HookHolder]
    ]
   ];
  HookHolder = Sort[HS];
  HS = HookHolder;
  Return[HS]]
For[i = 1, i < 31, i++,
For[j = 1, j < PartitionsP[i] + 1, j++,
  CurrSet=HookSet[Partitions[i][[j]]];
  If[CurrSet == Table[i,{i,1,Length[CurrSet]}],
   SGFHolder = SegGenFn + q^i;
   SegGenFn = SGFHolder]
  ]
]
(* second program: *)
h[l_] := Module[{n, s}, n = Length[l]; s = Table[Table[1 + l[[i]] - j + Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}] // Flatten // Union; If[s == Range[Max[Append[s, 0]]], 1, 0]]; g[n_, i_, l_] := g[n, i, l] = If[n == 0 || i == 1, h[Join[l, Array[1&, n]]], If[i<1, 0, g[n, i-1, l] + If[i>n, 0, g[n-i, i, Append[l, i]]]]]; a[n_] := g[n, n, {}]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 60}] (* Jean-François Alcover, Jan 22 2016, after Alois P. Heinz *)

a(31)-a(53) from Alois P. Heinz, Aug 12 2013

A224338 Number of idempotent 7 X 7 0..n matrices of rank 6.

Original entry on oeis.org

889, 10199, 57337, 218743, 653177, 1647079, 3670009, 7440167, 13999993, 24801847, 41803769, 67575319, 105413497, 159468743, 234881017, 337925959, 476171129, 658642327, 895999993, 1200725687, 1587318649, 2072502439, 2675441657

Offset: 1

R. H. Hardin, formula via M. F. Hasler, William J. Keith, and Rob Pratt in the Sequence Fans Mailing List, Apr 03 2013

Row 7 of A224333.

			Some solutions for n=1
..1..0..0..0..0..0..0....1..0..0..0..0..0..0....1..0..0..1..0..0..0
..0..1..0..0..0..0..0....0..1..0..0..0..1..0....0..1..0..0..0..0..0
..0..0..1..0..0..0..0....0..0..1..0..0..1..0....0..0..1..0..0..0..0
..0..0..0..0..0..1..1....0..0..0..1..0..0..0....0..0..0..0..0..0..0
..0..0..0..0..1..0..0....0..0..0..0..1..1..0....0..0..0..1..1..0..0
..0..0..0..0..0..1..0....0..0..0..0..0..0..0....0..0..0..0..0..1..0
..0..0..0..0..0..0..1....0..0..0..0..0..1..1....0..0..0..1..0..0..1

R. H. Hardin, Table of n, a(n) for n = 1..210
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A224333.

Vec(-7*x*(x^6-8*x^5+29*x^4+64*x^3+659*x^2+568*x+127)/(x-1)^7 + O(x^100)) \\ Colin Barker, Sep 20 2014

a(n) = 14*n^6 + 84*n^5 + 210*n^4 + 280*n^3 + 210*n^2 + 84*n + 7.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7). - Colin Barker, Sep 20 2014
G.f.: -7*x*(x^6-8*x^5+29*x^4+64*x^3+659*x^2+568*x+127) / (x-1)^7. - Colin Barker, Sep 20 2014

cp's OEIS Frontend

User: William J. Keith

A366920 a(n) is the number times a Dyck path in an m X m box of any size has area n, counted to the lower right.

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A298411 Coefficients of q^(-1/24)*eta(4q)^(1/2).

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A253926 a(n) is the excess of the number of Collatz permutations of length n (with first index 15) over the n-th Fibonacci number.

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A242168 Decimal expansion of the integral of the q-Pochhammer symbol (reciprocal of the partition function) over the real interval -1 to 1.

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Maple

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A239445 Values n at which ratios of successive partition numbers approach 1 closer than the reciprocal of a whole number.

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Mathematica

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A238752 Number of nonisomorphic partial 1-differential posets up to rank n.

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A236680 Dimension of the space of spinors in n-dimensional real space.

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Mathematica

PARI

Formula

A234937 Triangle read by rows of coefficients of polynomials generated by the Han/Nekrasov-Okounkov formula.

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