A047656 -id:A047656 - cp's OEIS frontend

A109354 a(n) = 6^((n^2 - n)/2).

1, 1, 6, 216, 46656, 60466176, 470184984576, 21936950640377856, 6140942214464815497216, 10314424798490535546171949056, 103945637534048876111514866313854976, 6285195213566005335561053533150026217291776, 2280250319867037997421842330085227917956272625811456

Offset: 0

Philippe Deléham, Aug 25 2005

nonn

Sequence given by the Hankel transform (see A001906 for definition) of A078018 = {1, 1, 7, 55, 469, 4237, 39907, 387739, ...}; example: det([1, 1, 7, 55; 1, 7, 55, 469; 7, 55, 469, 4237; 55, 469, 4237, 39907]) = 6^6 = 46656.

In general, sequences of the form m^((n^2 - n)/2) enumerate the graphs with n labeled nodes with m types of edge. a(n) therefore is the number of labeled graphs with n nodes with 6 types of edge. - Mark Stander, Apr 11 2019

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A000400, A006125, A047656, A053763, A053764, A109345.

Table[6^((n^2-n)/2),{n,0,10}] (* Harvey P. Dale, May 28 2013 *)

a(n) = 6^((n^2 - n)/2); \\ Michel Marcus, Apr 12 2019

a(n+1) is the determinant of n X n matrix M_(i, j) = binomial(6i, j).

G.f. A(x) satisfies: A(x) = 1 + x * A(6*x). - Ilya Gutkovskiy, Jun 04 2020

Terms a(11) and beyond from Andrew Howroyd, Jan 02 2020

A128386 Expansion of c(3x^2)/(1-xc(3*x^2)), c(x) the g.f. of A000108.

Original entry on oeis.org

1, 1, 4, 7, 28, 58, 232, 523, 2092, 4966, 19864, 48838, 195352, 492724, 1970896, 5068915, 20275660, 52955950, 211823800, 560198962, 2240795848, 5987822380, 23951289520, 64563867454, 258255469816, 701383563388, 2805534253552

Offset: 0

Paul Barry, Feb 28 2007

Hankel transform is 3^C(n+1,2) = A047656(n+1).

Series reversion of x*(1+x)/(1+2*x+4*x^2).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Alin Bostan, Computer Algebra for Lattice Path Combinatorics, Slides, Séminaire de Combinatoire Ph. Flajolet, March 28 2013.
Alin Bostan, Andrew Elvey Price, Anthony John Guttmann, and Jean-Marie Maillard, Stieltjes moment sequences for pattern-avoiding permutations, arXiv:2001.00393 [math.CO], 2020.

Cf. A000108, A009766, A047656, A120730.

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (Sqrt(1-12*x^2)+2*x-1)/(2*x*(1-4*x)) )); // G. C. Greubel, Nov 07 2022

A120730[n_, k_]:= If[n>2*k, 0, Binomial[n,k]*(2*k-n+1)/(k+1)];
A126386[n_]:= Sum[3^k*A120730[n, n-k], {k,0,n}];
Table[A126386[n], {n,0,50}] (* G. C. Greubel, Nov 07 2022 *)

def A120730(n, k): return 0 if (n>2*k) else binomial(n, k)*(2*k-n+1)/(k+1)
def A126386(n): return sum(3^k*A120730(n,n-k) for k in range(n+1))
[A126386(n) for n in range(51)] # G. C. Greubel, Nov 07 2022

G.f.: (sqrt(1-12*x^2)+2*x-1)/(2*x*(1-4*x)).
a(n) = (1/(n+1))*Sum_{k=0..n+1} Sum_{j=0..k} C(n,k)*C(k,j)*C(2*n-2*k+j, n-2*k+j)*(-1)^(n+j)*2^(2*k-j).
a(n) = Sum_{k=0..floor(n/2)} C(n,n-k)*(n-2*k+1)*3^k/(n-k+1);
a(n) = Sum_{k=0..floor(n/2)} A009766(n-k,k)*3^k.
a(n) = Sum_{k=0..n} 3^k*A120730(n,n-k). - Philippe Deléham, Mar 03 2007
D-finite with recurrence (n+1)*a(n) - 4*(n+1)*a(n-1) + 12*(2-n)*a(n-2) + 48*(n-2)*a(n-3) = 0. - R. J. Mathar, Nov 14 2011

A157783 Triangle read by rows: the coefficient [x^k] of the polynomial Product_{i=1..n} (3^(i-1)-x) in row n, column k, 0 <= k <= n.

Original entry on oeis.org

1, 1, -1, 3, -4, 1, 27, -39, 13, -1, 729, -1080, 390, -40, 1, 59049, -88209, 32670, -3630, 121, -1, 14348907, -21493836, 8027019, -914760, 33033, -364, 1, 10460353203, -15683355351, 5873190687, -674887059, 24995817, -298389, 1093

Offset: 0

Roger L. Bagula, Mar 06 2009

Row sums except n=0 are zero.

Triangle T(n,k), 0 <= k <= n, read by rows given by [1,q-1,q^2,q^3-q,q^4,q^5-q^2,q^6,q^7-q^3,q^8,...] DELTA [ -1,0,-q,0,-q^2,0,-q^3,0,-q^4,0,...] (for q=3)=[1,2,9,24,81,234,729,2160,6561,...] DELTA [ -1,0,-3,0,-9,0,-27,0,-81,0,-243,0,...] where DELTA is the operator defined in A084938; see A122006 and A000244. - Philippe Deléham, Mar 09 2009

			Triangle begins
  1;
  1, -1;
  3, -4, 1;
  27, -39, 13, -1;
  729, -1080, 390, -40, 1;
  59049, -88209, 32670, -3630, 121, -1;
  14348907, -21493836, 8027019, -914760, 33033, -364, 1;
  10460353203, -15683355351, 5873190687, -674887059, 24995817, -298389, 1093, -1;
  22876792454961, -34309958505840, 12860351387820, -1481851188720, 55340738838, -677572560, 2688780, -3280, 1;
Row n=3 is 27 - 39*x + 13*x^2 - x^3.

Cf. A157832, A135950, A022166, A047656 (column k=1), A003462 (subdiagonal k=n-1), A203243 (subdiagonal k=n-2).

A157783 := proc(n,k)
    product( 3^(i-1)-x,i=1..n) ;
    coeftayl(%,x=0,k) ;
end proc: # R. J. Mathar, Oct 15 2013

Clear[f, q, M, n, m];
q = 3;
f[k_, m_] := If[k == m, q^(n - k), If[m == 1 && k < n, q^(n - k), If[k == n && m == 1, -(n-1), If[k == n && m > 1, 1, 0]]]];
M[n_] := Table[f[k, m], {k, 1, n}, {m, 1, n}];
Table[M[n], {n, 1, 10}];
Join[{1}, Table[Expand[CharacteristicPolynomial[M[n], x]], {n, 1, 7}]];
a = Join[{{ 1}}, Table[CoefficientList[CharacteristicPolynomial[M[n], x], x], {n, 1, 7}]];
Flatten[a]

A109493 a(n) = 7^((n^2 - n)/2).

Original entry on oeis.org

1, 1, 7, 343, 117649, 282475249, 4747561509943, 558545864083284007, 459986536544739960976801, 2651730845859653471779023381601, 107006904423598033356356300384937784807

Offset: 0

Philippe Deléham, Aug 29 2005

Sequence given by the Hankel transform (see A001906 for definition) of A081178 = {1, 1, 8, 71, 680, 6882, 72528, 788019, ...}; example: det([1, 1, 8, 71; 1, 8, 71, 680; 8, 71, 680, 6882; 71, 680, 6882, 72528]) = 7^6 = 117649.

In general, sequences of the form m^((n^2 - n)/2) enumerate the graphs with n labeled nodes with m types of edge. a(n) therefore is the number of labeled graphs with n nodes with 7 types of edge. - Mark Stander, Apr 11 2019

Cf. A006125, A047656, A053763, A053764, A109345, A109354.

a(n) = 7^binomial(n,2) \\ Charles R Greathouse IV, Jan 17 2012

a(n+1) is the determinant of n X n matrix M_(i, j) = binomial(7i, j).

G.f. A(x) satisfies: A(x) = 1 + x * A(7*x). - Ilya Gutkovskiy, Jun 04 2020

A203148 (n-1)-st elementary symmetric function of {3,9,...,3^n}.

Original entry on oeis.org

1, 12, 351, 29160, 7144929, 5223002148, 11433166050879, 75035879252272080, 1477081305957768349761, 87223128348206814118735932, 15451489966710801620870785316511, 8211586182553137756809552940033725880, 13091937140529934508508023103481190655434529

Offset: 1

Clark Kimberling, Dec 29 2011

nonn

From R. J. Mathar, Oct 01 2016: (Start)
The k-th elementary symmetric functions of the integers 3^j, j=1..n, form a triangle T(n,k), 0<=k<=n, n>=0:
  1;
  1   3;
  1  12    27;
  1  39   351    729;
  1 120  3510  29160   59049;
  1 363 32670 882090 7144929 14348907;
which is the row-reversed version of A173007. This here is the first subdiagonal. The diagonal seems to be A047656. The first column is A029858. (End)

G. C. Greubel, Table of n, a(n) for n = 1..60

Cf. A203149.

Cf. A029858, A047656, A173007.

[(1/2)*(3^n -1)*3^(Binomial(n,2)): n in [1..20]]; // G. C. Greubel, Feb 24 2021

f[k_]:= 3^k; t[n_]:= Table[f[k], {k, 1, n}];
a[n_]:= SymmetricPolynomial[n - 1, t[n]];
Table[a[n], {n, 1, 16}] (* A203148 *)
Table[1/2 (3^n - 1) 3^Binomial[n, 2], {n, 1, 20}] (* Emanuele Munarini, Sep 14 2017 *)

[(1/2)*(3^n -1)*3^(binomial(n,2)) for n in (1..20)] # G. C. Greubel, Feb 24 2021

a(n) = (1/2)*(3^n-1)*3^(binomial(n,2)). - Emanuele Munarini, Sep 14 2017

A290000 a(n) = Product_{k=1..n-1} (3^k + 1).

Original entry on oeis.org

1, 1, 4, 40, 1120, 91840, 22408960, 16358540800, 35792487270400, 234870301468364800, 4623187014103292723200, 272999193182799435304960000, 48361261073946554365403054080000, 25701205307660304745058529866383360000, 40976048450930207702360695570691784048640000

Offset: 0

Ilya Gutkovskiy, Jun 06 2020

nonn

G. C. Greubel, Table of n, a(n) for n = 0..50

Cf. A027871, A034472, A047656, A119600, A132324, A323716.

Sequences of the form Product_{j=1..n-1} (m^j + 1): A000012 (m=0), A011782 (m=1), A028362 (m=2), this sequence (m=3), A309327 (m=4).

[n lt 3 select 1 else (&*[3^j +1: j in [1..n-1]]): n in [1..20]]; // G. C. Greubel, Feb 21 2021

Table[Product[3^k + 1, {k, 1, n - 1}], {n, 0, 14}]

a(n) = prod(k=1, n-1, 3^k + 1); \\ Michel Marcus, Jun 06 2020

from sage.combinat.q_analogues import q_pochhammer
[1]+[3^(binomial(n,2))*q_pochhammer(n-1, -1/3, 1/3) for n in (1..20)] # G. C. Greubel, Feb 21 2021

G.f. A(x) satisfies: A(x) = 1 + x * A(3*x) / (1 - x).
G.f.: Sum_{k>=0} 3^(k*(k - 1)/2) * x^k / Product_{j=0..k-1} (1 - 3^j*x).
a(0) = 1; a(n) = Sum_{k=0..n-1} 3^k * a(k).
a(n) ~ c * 3^(n*(n - 1)/2), where c = Product_{k>=1} (1 + 1/3^k) = 1.564934018567011537938849... = A132324.
a(n) = 3^(binomial(n+1,2))*(-1/3;1/3){n}, where (a;q){n} is the q-Pochhammer symbol. - G. C. Greubel, Feb 21 2021

A109966 a(n) = 8^((n^2-n)/2).

Original entry on oeis.org

1, 1, 8, 512, 262144, 1073741824, 35184372088832, 9223372036854775808, 19342813113834066795298816, 324518553658426726783156020576256, 43556142965880123323311949751266331066368, 46768052394588893382517914646921056628989841375232, 401734511064747568885490523085290650630550748445698208825344

Offset: 0

Philippe Deléham, Sep 01 2005

Sequence given by the Hankel transform (see A001906 for definition) of A082147 = {1, 1, 9, 89, 945, 10577, 123129, 1476841, ...}; example: det([1, 1, 9, 89; 1, 9, 89, 945; 9, 89, 945, 10577; 89, 945, 10577, 123129]) = 8^6 = 262144.

The number of labeled multigraphs on n vertices such that (i) no self loops are allowed; (ii) all edges are painted in one of 3 colors; (iii) edges between any pair of vertices are painted in distinct colors. Note, this implies that there are at most 3 edges between any vertex pair. Also note there is no restriction on the color of edges incident to a common vertex. - Geoffrey Critzer, Jan 14 2020

Cf. A006125, A047656, A053763, A053764, A059435, A082147, A109345, A109354, A109493.

[2^(3*Binomial(n,2)): n in [0..10]]; // G. C. Greubel, Feb 05 2018

Table[2^(3*Binomial[n,2]),{n,0,10}] (* Geoffrey Critzer, Nov 10 2011 *)

a(n)=8^binomial(n,2) \\ Charles R Greathouse IV, Jan 17 2012

a(n+1) is the determinant of n X n matrix M_(i, j) = binomial(8i, j).

Hankel transform of A059435. - Philippe Deléham, Sep 03 2006

a(10) corrected and a(11), a(12) from Georg Fischer, Apr 01 2022

A263394 a(n) = Product_{i=1..n} (3^i - 2^i).

Original entry on oeis.org

1, 5, 95, 6175, 1302925, 866445125, 1784010512375, 11248186280524375, 215638979183932793125, 12512451767147700321078125, 2190917791975795178520458609375, 1155369543009475708416871245360859375, 1832567448623162714866960405275465241328125

Offset: 1

Bob Selcoe, Mar 02 2016

Generally, for sequences of the form a(n) = Product_{i=1..n} j^i-k^i, where j>k>=1 and n>=1: given probability p=(k/j)^n that an outcome will occur at the n-th stage of an infinite process, then r = 1 - a(n)/j^((n^2+n)/2) is the probability that the outcome has occurred up to and including the n-th iteration. Here, j=3 and k=2, so p=(2/3)^n and r = 1-a(n)/A047656(n+1). The limiting ratio of r ~ 0.9307279.

Cf. A001047, A047656.
Cf. sequences of the form Product_{i=1..n}(j^i - 1): A005329 (j=2), A027871 (j=3), A027637 (j=4), A027872 (j=5), A027873 (j=6), A027875 (j=7),A027876 (j=8), A027877 (j=9), A027878 (j=10), A027879 (j=11), A027880 (j=12).
Cf. sequences of the form Product_{i=1..n}(j^i - k^1), k>1: A269576 (j=4, k=3), A269661 (j=5, k=4).

[&*[ 3^k-2^k: k in [1..n] ]: n in [1..16]]; // Vincenzo Librandi, Mar 03 2016

A263394:=n->mul(3^i-2^i, i=1..n): seq(A263394(n), n=1..15); # Wesley Ivan Hurt, Mar 02 2016

Table[Product[3^i - 2^i, {i, n}], {n, 15}] (* Wesley Ivan Hurt, Mar 02 2016 *)
FoldList[Times,Table[3^i-2^i,{i,15}]] (* Harvey P. Dale, Feb 06 2017 *)

a(n) = prod(k=1, n, 3^k-2^k); \\ Michel Marcus, Mar 05 2016

a(n) = Product_{i=1..n} A001047(i).

a(n) ~ c * 3^(n*(n+1)/2), where c = QPochhammer(2/3) = 0.0692720728018644... . - Vaclav Kotesovec, Oct 10 2016

A294352 Product of first n terms of the binomial transform of the factorial.

Original entry on oeis.org

1, 2, 10, 160, 10400, 3390400, 6635012800, 90899675360000, 9962695319131360000, 9827302289744364817600000, 96937502343569678741652977600000, 10518214548789290471667075399621491200000, 13695360582395151673134516587047571322777664000000

Offset: 0

Vaclav Kotesovec, Oct 29 2017

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..43
M. Bahrami-Taghanaki, A. R. Moghaddamfar, Nima Salehy, and Navid Salehy, Some Identities Involving Stirling Numbers Arising from Matrix Decompositions, J. Int. Seq. (2024) Vol. 24, No. 5, Art. No. 24.5.3. See p. 9.

Cf. A000522, A047656, A086619, A294353.

Table[Product[Sum[Binomial[m, k]*k!, {k, 0, m}], {m, 0, n}], {n, 0, 12}]

a(n) ~ c * exp(n+1) * BarnesG(n+2).
a(n) ~ c * n^(n^2/2 + n + 5/12) * (2*Pi)^(n/2 + 1/2) / (A * exp(3*n^2/4 - 13/12))
where c = 0.24314714161123874545254157058990661627416712475691705561000082745...
and A is the Glaisher-Kinkelin constant A074962.

A042891 Denominators of continued fraction convergents to sqrt(977).

Original entry on oeis.org

1, 3, 4, 35, 179, 214, 393, 1393, 10144, 11537, 21681, 163304, 511593, 674897, 1186490, 6607347, 54045266, 60652613, 236003105, 14692845123, 44314538474, 59007383597, 516373607250, 2640875419847, 3157249027097

Offset: 0

N. J. A. Sloane

The Hankel transform (see A001906 for definition) of this sequence forms the sequence A047656(n+1)= [1,3,27,729,59049,14348907,...] . - Philippe Deléham, Aug 29 2006

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 14753497736, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).

Cf. A042890, A040945.

Denominator[Convergents[Sqrt[977], 40]] (* Harvey P. Dale, May 09 2012 *)

The Hankel transform (see A001906 for definition) of this sequence forms the sequence A047656(n+1)= [1,3,27,729,59049,14348907,...]. - Philippe Deléham, Aug 29 2006

a(n) = 14753497736*a(n-19) - a(n-38) for n > 37. - Vincenzo Librandi, Jan 31 2014

cp's OEIS Frontend

A109354 a(n) = 6^((n^2 - n)/2).

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A128386 Expansion of c(3*x^2)/(1-x*c(3*x^2)), c(x) the g.f. of A000108.

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A157783 Triangle read by rows: the coefficient [x^k] of the polynomial Product_{i=1..n} (3^(i-1)-x) in row n, column k, 0 <= k <= n.

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A109493 a(n) = 7^((n^2 - n)/2).

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A203148 (n-1)-st elementary symmetric function of {3,9,...,3^n}.

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A290000 a(n) = Product_{k=1..n-1} (3^k + 1).

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A109966 a(n) = 8^((n^2-n)/2).

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A128386 Expansion of c(3x^2)/(1-xc(3*x^2)), c(x) the g.f. of A000108.