A048996 -id:A048996 - cp's OEIS frontend

A179313 Triangle T(n,k) read by rows: product of the compositorial weight of the k-th partition of n times A074664(.) applied to each part.

1, 1, 1, 2, 2, 1, 6, 4, 1, 3, 1, 22, 12, 4, 6, 3, 4, 1, 92, 44, 12, 4, 18, 12, 1, 8, 6, 5, 1, 426, 184, 44, 24, 66, 36, 12, 6, 24, 24, 4, 10, 10, 6, 1, 2146, 852, 184, 88, 36, 276, 132, 72, 18, 12, 88, 72, 24, 24, 1, 30, 40, 10, 12, 15, 7, 1, 11624, 4292, 852, 368, 264, 1278, 552

Offset: 1

Alford Arnold, Jul 11 2010

Row n has A000041(n) entries. T(n,k) is the product of A074664(a_i) over all parts a_i
multiplied by the compositorial weight A048996(n,k) of the k-th partition (Abramowitz-Stegun order)
of n = sum_i a_i.
Summing also over the partitions with a common number of parts would create A127743.
In row n=4, for example, the partitions 3+1 and 2+2, each with 2 parts, are represented by
T(4,2)=4 and T(4,3)= 1 here, and the sum 4+1=5 of the entries is the single entry A127743(4,.).
In this sense, the table is a refinement of A127743.

			T(6,3) represents the 3rd partition of 6, namely 2+4. A074664(2)*A074664(4) = 1*6 is multiplied
by the weight A048996([2,4]) = 2!/1!/1! =2, and T(6,3) =1*6*2=12.
T(6,5) represents the 5th partition of 6, namely 1+1+4. A074664(1)*A074664(1)*A074664(4) = 1*1*6 is multiplied
by the weight A048996([1,1,4]) = 3!/2!/1! =3, and T(6,5) =1*1*6*3.
T(7,6) represents the 6th partition of 7, namely 1+2+4. A074664(1)*A074664(2)*A074664(4) = 1*1*6 is multiplied
the weight A048996([1,2,4]) = 3!/1!/1!/1! =6, and T(7,6) =1*1*6*6.
The triangle starts
1;
1,1;
2,2,1;
6,4,1,3,1;
22,12,4,6,3,4,1;
92,44,12,4,18,12,1,8,6,5,1;
426,184,44,24,66,36,12,6,24,24,4,10,10,6,1;

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, p. 831

Cf. A000041, A127743

T(n,k) = A048996(n,k) * A179380(n,k).

sum_{k=1..A000041(n)} T(n,k) = A000110(n).

Edited and extended by R. J. Mathar, Jul 16 2010

A157163 Product_{n>=1} (1 + 2a(n)x^n) = Sum_{k>=0} binomial(2k, k)x^k = 1/sqrt(1 - 4*x), with the central binomial numbers A000984(n).

Original entry on oeis.org

1, 3, 4, 27, 48, 156, 576, 2955, 7168, 27792, 95232, 352188, 1290240, 5105856, 17743872, 77010795, 252641280, 1000224768, 3616800768, 14484040464, 52102692864, 208963943424, 764877471744, 3025006038012, 11258183024640, 44968060784640, 166308918329344

Offset: 1

Wolfdieter Lang, Aug 10 2009

In the original problem 2*a(n) = [2, 6, 8, 54, 96, 312, 1152, 5910, 14336, 55584, 190464, 704376, ...] appears.

			Recurrence I: a(4) = binomial(8, 4)/2 - 2*a(1)*a(3) = 35 - 8 = 27.
Recurrence II: a(4) = (1/2)*(1/2)*(-2*a(2))^2 + (1/2)*(1*cbi(4) - (1/2)*(2*cbi(1)*cbi(3) + 1*cbi(2)^2) + (1/3)*3*cbi(1)^2*cbi(2)) = 27.
Recurrence II (rewritten): a(4)= (1/8)*((-2)^4 + 2*(-2*a(2))^2 + (1/2)*4^4) = 27.

Robert Israel, Table of n, a(n) for n = 1..1665
Giedrius Alkauskas, One curious proof of Fermat's little theorem, arXiv:0801.0805 [math.NT], 2008.
Giedrius Alkauskas, A curious proof of Fermat's little theorem, Amer. Math. Monthly 116(4) (2009), 362-364.
Giedrius Alkauskas, Algebraic functions with Fermat property, eigenvalues of transfer operator and Riemann zeros, and other open problems, arXiv:1609.09842 [math.NT], 2016.
Art of Problem Solving, Product formula for a generating function
H. Gingold, H. W. Gould, and Michael E. Mays, Power Product Expansions, Utilitas Mathematica 34 (1988), 143-161.
H. Gingold and A. Knopfmacher, Analytic properties of power product expansions, Canad. J. Math. 47 (1995), 1219-1239.
W. Lang, Recurrences for the general problem.

Cf. A147542 (Fibonacci), A157161 (Catalan).

N:= 100: # for a(1)..a(N)
S:= convert(series(-ln(1-4*x)/2,x,N+1),polynom):
for n from 1 to N do
  a[n]:= coeff(S,x,n)/2;
  S:= S - add((-1)^(k-1)*(2*a[n])^k*x^(k*n)/k, k=1..N/n)
od:
seq(a[n],n=1..N); # Robert Israel, Jan 03 2019

a(n) = if (n==1, 1, (1/(2*n))*((-2*a(1))^n + sumdiv(n, d, if ((d!=1) && (d!=n), d*(-2*a(d))^(n/d), 0)) + 4^n/2)); \\ after 2nd Recurrence II; Michel Marcus, Jul 06 2015

Recurrence I: With FP(n,m) the set of partitions of n with m distinct parts (which could be called fermionic partitions fp):
a(n) = binomial(2*n, n)/2 - Sum_{m=2..maxm(n)} 2^(m-1)*(Sum_{fp from FP(n,m)} (Product_{j=1..m} a(k[j]))), with maxm(n) = A003056(n) and the distinct parts k[j], j = 1..m, of the partition fp of n, n >= 3. Inputs a(1) = 1, a(2) = 3. See the array A008289(n,m) for the cardinality of the set FP(n,m).
Recurrence II: With P(n,m) the set of all partitions of n with m parts, and the multinomial numbers M0 (given for every partition under A048996):
a(n) = (1/2)*Sum_{d|n, 1= 2; a(1) = 1, with cbi(n) = binomial(2*n, n) = A000984(n). The exponents e(j) >= 0 satisfy Sum_{j=1..n} j*e(j) =n and Sum_{j=1..n} e(j) = m. The M0 numbers are m!/(Product_{j=1..n} (e(j))!).
Recurrence II (rewritten, due to email from V. Jovovic, Mar 10 2009):
a(n) = ((-2*a(1))^n + Sum_{d|n, 1

A179233 Irregular triangle T(n,k) = A049019(n,k)/A096162(n,k) read along rows, 1<=k <= A000041(n).

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 8, 3, 18, 1, 1, 10, 20, 30, 45, 40, 1, 1, 12, 30, 10, 45, 360, 15, 80, 270, 75, 1, 1, 14, 42, 70, 63, 630, 210, 315, 140, 2520, 420, 175, 1050, 126, 1, 1, 16, 56, 112, 35, 84, 1008, 1680, 630, 840, 224, 5040, 1680

Offset: 1

Alford Arnold, Jul 08 2010

Each row n of A049019, of A096162 and of the triangle here has A000041(n) entries.

			A049019(.,.) begins 1; 1; 2, 1; 6, 6, 1; 8, 6, 36, 24, ...
A096162(.,.) begins 1; 1; 2, 1; 1, 6, 1; 1, 2, 2, 24 ...
so
T(.,.) begins ..... 1; 1; 1, 1; 6, 1, 1; 8, 3, 18, 1 ...

Cf. A000041, A000670, A096161.

T(n,k) = A049019(n,k) / A096162(n,k) = A048996(n,k) * A036040(n,k).

Sum_{k=1..A000041(n)} T(n,k) = A120774(n).

Extended, and bivariate indices restored - R. J. Mathar, Jul 13 2010

A119443 Distribution of A060642 in Abramowitz and Stegun order.

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 5, 6, 4, 6, 1, 7, 10, 12, 9, 12, 8, 1, 11, 14, 20, 9, 15, 36, 8, 12, 24, 10, 1, 15, 22, 28, 30, 21, 60, 27, 36, 20, 72, 32, 15, 40, 12, 1, 22, 30, 44, 42, 25, 33, 84, 90, 60, 54, 28, 120, 54, 144, 16, 25, 120, 80, 18, 60, 14, 1

Offset: 1

Alford Arnold, May 22 2006

			1;
2,1;
3,4,1;
5,6,4,6,1;
7,10,12,9,12,8,1;

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A000041 (row lengths), A048996, A119441.

a(n,k) = A119441(n,k) * A048996(n,k)

More terms from R. J. Mathar, Jul 12 2013

A322480 Irregular triangular array read by rows: T(n,k), n>=1, is the number of ordered factorizations corresponding to each unordered factorization, indexed by k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 1, 2, 2, 3, 1, 1, 2, 2, 3, 1, 2, 1, 2, 1, 1, 2, 2, 2, 3, 6, 4, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 3, 1, 1, 2, 2, 2, 6, 1, 1, 2, 2, 3, 3, 4, 1, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 3, 1, 6, 3, 6, 1, 1, 2, 1, 2, 1, 2, 2, 3, 2, 6, 4, 1, 1, 2, 2, 2, 6, 1, 1, 2, 2, 3, 1, 2, 2, 3, 1, 2, 1, 1, 2, 2, 2, 3, 2, 6, 6, 4, 12, 5

Offset: 1

Thomas Anton, Dec 09 2018

The method of indexing the unordered factorizations of n in this array is as follows: take all unordered factorizations of n and write them with their factors in nonincreasing order (e.g., 2*4*5*3 becomes 5*4*3*2), and order these reverse-lexicographically (e.g., for 12: 12, 6*2, 4*3, 3*2*2), then assign the index k to the k-th factorization in this ordering.
For a sequence f with Dirichlet inverse f^(-1), f^(-1)(n) is the sum over all multisets M of integers > 1 with product n, of the product of the terms f(m) with indices m in M (counted with multiplicity) multiplied by T(n,k)*(-1)^c/f(1)^(c+1) where c = |M| and T(n,k) corresponds to M.
The multiset of entries in the n-th row is determined by the prime signature of n.
For the p^j-th row with p a prime, the entries give the number of compositions of j corresponding to each partition of j, indexed by k in an analogous manner, given by the j-th row of A048996.

			  1;
  1;
  1;
  1, 1;
  1;
  1, 2;
  1;
  1, 2, 1;
  1, 1;
  1, 2;
  1;
  1, 2, 2, 3;
  etc.
The 12th row is 1,2,2,3, because 12 can be factored as 12, 6*2, 3*4 or 3*2*2 with respective sets of ordered factorizations {12}, {6*2, 2*6}, {4*3, 3*4} and {3*2*2, 2*3*2, 2*2*3}, with respective cardinalities 1, 2, 2 and 3.

Cf. A048996, A002033 (row sums), A212171, A251683, A001055 (row lengths).

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A179313 Triangle T(n,k) read by rows: product of the compositorial weight of the k-th partition of n times A074664(.) applied to each part.

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A179380 Triangle T(n,k) read by rows: product of A074664(a_i) of all parts a_i of the k-th partition of n.

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A157163 Product_{n>=1} (1 + 2*a(n)*x^n) = Sum_{k>=0} binomial(2*k, k)*x^k = 1/sqrt(1 - 4*x), with the central binomial numbers A000984(n).

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A179233 Irregular triangle T(n,k) = A049019(n,k)/A096162(n,k) read along rows, 1<=k <= A000041(n).

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A119443 Distribution of A060642 in Abramowitz and Stegun order.

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A322480 Irregular triangular array read by rows: T(n,k), n>=1, is the number of ordered factorizations corresponding to each unordered factorization, indexed by k.

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A157163 Product_{n>=1} (1 + 2a(n)x^n) = Sum_{k>=0} binomial(2k, k)x^k = 1/sqrt(1 - 4*x), with the central binomial numbers A000984(n).