A257565 -id:A257565 - cp's OEIS frontend

A003154 Centered 12-gonal numbers, or centered dodecagonal numbers: numbers of the form 6k(k-1) + 1.

1, 13, 37, 73, 121, 181, 253, 337, 433, 541, 661, 793, 937, 1093, 1261, 1441, 1633, 1837, 2053, 2281, 2521, 2773, 3037, 3313, 3601, 3901, 4213, 4537, 4873, 5221, 5581, 5953, 6337, 6733, 7141, 7561, 7993, 8437, 8893, 9361, 9841, 10333, 10837, 11353, 11881, 12421

Offset: 1

N. J. A. Sloane

Binomial transform of [1, 12, 12, 0, 0, 0, ...]. Narayana transform (A001263) of [1, 12, 0, 0, 0, ...]. - Gary W. Adamson, Dec 29 2007
Numbers k such that 6*k+3 is a square, these squares are given in A016946. - Gary Detlefs and Vincenzo Librandi, Aug 08 2010
Odd numbers of the form floor(n^2/6). - Juri-Stepan Gerasimov, Jul 27 2011
Bisection of A032528. - Omar E. Pol, Aug 20 2011
Sequence found by reading the line from 1, in the direction 1, 13, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. Opposite numbers to the members of A033581 in the same spiral. - Omar E. Pol, Sep 08 2011
The digital root has period 3 (1, 4, 1) (A146325), the same digital root as the centered triangular numbers A005448(n). - Peter M. Chema, Dec 20 2023

			From _Omar E. Pol_, Aug 21 2011: (Start)
1. Classic illustration of initial terms of the star numbers:
.
.                                     o
.                                    o o
.                  o            o o o o o o o
.               o o o o          o o o o o o
.     o          o o o            o o o o o
.               o o o o          o o o o o o
.                  o            o o o o o o o
.                                    o o
.                                     o
.
.     1            13                 37
.
2. Alternative illustration of initial terms using n-1 concentric hexagons around a central element:
.
.                                 o o o o o
.                                o         o
.                o o o          o   o o o   o
.               o     o        o   o     o   o
.     o        o   o   o      o   o   o   o   o
.               o     o        o   o     o   o
.                o o o          o   o o o   o
.                                o         o
.                                 o o o o o
(End)

Martin Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, p. 20.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1000
John Elias, Illustration: Star Configurations on the Zero-Centered Hexagonal Number Spiral.
John Elias, Illustration: Star Configurations on the Zero-Centered Square and Hexagonal Number Spirals.
John Elias, Illustration: Generalized Pentagonal and Octagonal Numbers in the Star-Crossed Configurations.
John Elias, Illustration: Generalized Pentagonal and Octagonal Integration in Centered 9-gonal Triangles.
Martin Gardner and N. J. A. Sloane, Correspondence, 1973-74.
Marco Matone and Roberto Volpato, Vector-Valued Modular Forms from the Mumford Form, Schottky-Igusa Form, Product of Thetanullwerte and the Amazing Klein Formula, arXiv:1102.0006 [math.AG], 2011-2012, c_n.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Amelia C. Sparavigna, Groupoid of OEIS A003154 numbers (star numbers or centered dodecagonal numbers), Politecnico di Torino, Repository istituzionale (2019).
Amelia Carolina Sparavigna, Groupoid of OEIS A003154 Numbers (star numbers or centered dodecagonal numbers), Department of Applied Science and Technology, Politecnico di Torino (Italy, 2019).
Amelia Carolina Sparavigna, Generalized Sum of Stella Octangula Numbers, Politecnico di Torino (Italy, 2021).
Leo Tavares, Illustration: Twin Hexagons.
Leo Tavares, Illustration: Diamond Rays.
Eric Weisstein's World of Mathematics, Star Number.
R. Yin, J. Mu, and T. Komatsu, The p-Frobenius Number for the Triple of the Generalized Star Numbers, Preprints 2024, 2024072280. See p. 1.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
Index entries for sequences related to centered polygonal numbers.

Cf. A001263, A001318, A003215, A007588, A016946, A032528, A033581, A049598, A056827, A306980.
Row 4 of A257565.
Cf. A000217, A005448, A016754, A146325.

List([1..50], n-> 12*Binomial(n,2)+1 ); # G. C. Greubel, Jul 23 2019

([: >: 6 * ] * <:) i.1000 NB. Stephen Makdisi, May 06 2018

[12*Binomial(n,2)+1: n in [1..50]]; // G. C. Greubel, Jul 23 2019

A003154:=n->6*n*(n-1) + 1: seq(A003154(n), n=1..100); # Wesley Ivan Hurt, Oct 23 2017

FoldList[#1 + #2 &, 1, 12 Range@50] (* Robert G. Wilson v *)
LinearRecurrence[{3,-3,1},{1,13,37},50] (* Harvey P. Dale, Jul 18 2016 *)
12*Binomial[Range[50], 2] + 1 (* G. C. Greubel, Jul 23 2019 *)

a(n)=6*n*(n-1)+1 \\ Charles R Greathouse IV, Nov 20 2012

print([6*n*(n-1)+1 for n in range(1, 47)]) # Michael S. Branicky, Jan 13 2021

G.f.: x*(1+10*x+x^2)/(1-x)^3. Simon Plouffe in his 1992 dissertation
a(n) = 1 + Sum_{j=0..n} (12*j). E.g., a(2)=37 because 1 + 12*0 + 12*1 + 12*2 = 37. - Xavier Acloque, Oct 06 2003
a(n) = numerator in B_2(x) = (1/2)x^2 - (1/2)x + 1/12 = Bernoulli polynomial of degree 2. - Gary W. Adamson, May 30 2005
a(n) = 12*(n-1) + a(n-1), with n>1, a(1)=1. - Vincenzo Librandi, Aug 08 2010
a(n) = A049598(n-1) + 1. - Omar E. Pol, Oct 03 2011
Sum_{n>=1} 1/a(n) = A306980 = Pi * tan(Pi/(2*sqrt(3))) / (2*sqrt(3)). - Vaclav Kotesovec, Jul 23 2019
From Amiram Eldar, Jun 21 2020: (Start)
Sum_{n>=1} a(n)/n! = 7*e - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 7/e - 1. (End)
a(n) = 2*A003215(n-1) - 1. - Leo Tavares, Jul 30 2021
E.g.f.: exp(x)*(1 + 6*x^2) - 1. - Stefano Spezia, Aug 19 2022

More terms from Michael Somos

A321189 a(n) = n! * [x^n] 1 - 1/(n - 1/(exp(x) - 1)).

Original entry on oeis.org

1, 1, 5, 73, 2169, 108901, 8288293, 890380177, 128364028145, 23918924529901, 5595490598128221, 1605718043992482553, 554663179293965398825, 227038711419826844827381, 108674023653792712066606229, 60142879347501714200454327841, 38108071228342727619600464659425

Offset: 0

Ilya Gutkovskiy, Oct 29 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..200
Robert Gill, The number of elements in a generalized partition semilattice, Discrete mathematics 186.1-3 (1998): 125-134. See Example 1.

Main diagonal of A257565.

Cf. A000670, A050351, A050352, A050353, A094420.

Concatenation([1],List([1..16],n->Sum([1..n],k->Stirling2(n,k)*Factorial(k)*n^(k-1)))); # Muniru A Asiru, Oct 29 2018

seq(coeff(series(factorial(n)*(1-1/(n-1/(exp(x)-1))),x,n+1), x, n), n = 0 .. 15); # Muniru A Asiru, Oct 29 2018
# Or, using the recurrence of the Fubini polynomials:
F := proc(n) option remember; if n = 0 then return 1 fi;
expand(add(binomial(n, k)*F(n-k)*x, k = 1..n)) end:
a := n -> `if`(n=0, 1, subs(x = n, F(n)) / n):
seq(a(n), n = 0..16);  # Peter Luschny, May 21 2021

Table[n! SeriesCoefficient[1 - 1/(n - 1/(Exp[x] - 1)), {x, 0, n}], {n, 0, 16}]
Join[{1}, Table[Sum[StirlingS2[n, k] k! n^(k - 1), {k, n}], {n, 16}]]

{a(n) = if(n==0, 1, sum(k=0, n, k!*n^(k-1)*stirling(n, k, 2)))} \\ Seiichi Manyama, Jun 12 2020

a(0) = 1; a(n) = Sum_{k=1..n} Stirling2(n, k)*k!*n^(k-1).
a(n) = A257565(n, n).
From Vaclav Kotesovec, Oct 29 2018: (Start)
a(n) ~ exp(1/2) * n! * n^(n-1).
a(n) ~ sqrt(2*Pi) * n^(2*n - 1/2) / exp(n - 1/2). (End)
a(n) = F_{n}(n) / n for n >= 1, where F_{n}(x) is the Fubini polynomial. In other words: a(n) = A094420(n) / n for n >= 1. - Peter Luschny, May 21 2021

A338875 Array T(n, m) read by ascending antidiagonals: numerators of shifted Fubini numbers F(n, m) where m >= 0.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 13, 5, 1, 1, 75, 2, 5, 1, 1, 541, 191, 29, 29, 1, 1, 4683, 76, 263, 149, 7, 1, 1, 47293, 5081, 4157, 24967, 2687, 727, 1, 1, 545835, 674, 93881, 115567, 44027, 66247, 631, 1, 1, 7087261, 386237, 21209, 377909, 31627, 37728769, 354061, 4481, 1, 1

Offset: 0

Stefano Spezia, Dec 25 2020

			Array T(n, m):
n\m|   0       1       2       3 ...
---+--------------------------------
0  |   1       1       1       1 ...
1  |   1       1       1       1 ...
2  |   3       5       5      29 ...
3  |  13       2      29     149 ...
...
Related table of shifted Fubini numbers F(n, m):
   1   1      1         1 ...
   1 1/2    1/6      1/24 ...
   3 5/6   5/36   29/1440 ...
  13   2 29/180 149/11520 ...
  ...

Takao Komatsu, Shifted Bernoulli numbers and shifted Fubini numbers, Linear and Nonlinear Analysis, Volume 6, Number 2, 2020, 245-263.

Cf. A000012 (n = 0 and n = 1), A000670 (m = 0), A226513 (high-order Fubini numbers), A232472, A232473, A232474, A257565, A338873, A338874.

Cf. A338876 (denominators).

F[n_,m_]:=n!Coefficient[Series[x^m/(x^m-Exp[x]+Sum[x^k/k!,{k,0,m}]),{x,0,n}],x,n]; Table[Numerator[F[n-m,m]],{n,0,9},{m,0,n}]//Flatten

tm(n, m) = {my(m = matrix(n, n, i, j, if (i==1, if (j==1, 1/(m + 1)!, if (j==2, 1)), if (j==1, (-1)^(i+1)/(m + i)!)))); for (i=2, n, for (j=2, n, m[i, j] = m[i-1, j-1]; ); ); m; }
T(n, m) = numerator(n!*matdet(tm(n, m))); \\ Michel Marcus, Dec 31 2020

T(n, m) = numerator(F(n, m)).
F(n, m) = n!*det(M(n, m)) where M(n, m) is the n X n Toeplitz matrix whose first row consists in 1/(m + 1)!, 1, 0, ..., 0 and whose first column consists in 1/(m + 1)!, -1/(m + 2)!, ..., (-1)^(n-1)/(m + n)! (see Proposition 5.1 in Komatsu).
F(n, m) = n!*Sum_{k=0..n-1} F(k, m)/((n - k + m)!*k!) for n > 0 and m >= 0 with F(0, m) = 1 (see Lemma 5.2).
F(n, m) = [x^n] n!*x^m/(x^m - exp(x) + E_m(x)), where E_m(x) = Sum_{n=0..m} x^n/n! (see Theorem 5.3 in Komatsu).
F(n, m) = n!*Sum_{k=1..n} Sum_{i_1+...+i_k=n, i_1,...,i_k>=1} Product_{j=1..k} 1/(i_j + m)! for n > 0 and m >= 0 (see Theorem 5.4).
F(1, m) = 1/(m + 1)! (see Theorem 5.5 in Komatsu).
F(n, m) = n!*Sum_{t_1+2*t_2+...+n*t_n=n} (t_1,...,t_n)!*(-1)^(n-t_1-...-t_n)*Product_{j=1..n} (1/(m + j)!)^t_j for n >= m >= 1 (see Theorem 5.7 in Komatsu).
(-1)^(n-1)/(n + m)! = det(M(n, m)) where M(n, m) is the n X n Toeplitz matrix whose first row consists in F(1, m), 1, 0, ..., 0 and whose first column consists in F(1, m), F(2, m)/2!, ..., F(n, m)/n! for n > 0 (see Theorem 5.8 in Komatsu).
Sum_{k=0..n} binomial(n, k)*F(k, m)*F(n-k, m) = - n!/(m^2*m!)*Sum_{l=0..n-1} ((m! + 1)/(m*m!))^(n-l-1)*(l*(m! + 1) - m)/l!*F(l, m) - (n - m)/m*F(n, m) for m > 0 (see Theorem 5.11 in Komatsu).

A338876 Array T(n, m) read by ascending antidiagonals: denominators of shifted Fubini numbers F(n, m) where m >= 0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 6, 6, 1, 1, 1, 36, 24, 1, 1, 30, 180, 1440, 120, 1, 1, 3, 1080, 11520, 2400, 720, 1, 1, 42, 9072, 2419200, 2016000, 1814400, 5040, 1, 1, 1, 90720, 11612160, 60480000, 435456000, 12700800, 40320, 1, 1, 90, 7776, 33177600, 69120000, 548674560000, 21337344000, 812851200, 362880, 1

Offset: 0

Stefano Spezia, Dec 25 2020

			Array T(n, m):
n\m|   0       1       2       3 ...
---+--------------------------------
0  |   1       1       1       1 ...
1  |   1       2       6      24 ...
2  |   1       6      36    1440 ...
3  |   1       1     180   11520 ...
...
Related table of shifted Fubini numbers F(n, m):
   1   1      1         1 ...
   1 1/2    1/6      1/24 ...
   3 5/6   5/36   29/1440 ...
  13   2 29/180 149/11520 ...
  ...

Takao Komatsu, Shifted Bernoulli numbers and shifted Fubini numbers, Linear and Nonlinear Analysis, Volume 6, Number 2, 2020, 245-263.

Cf. A000012 (n = 0 or m = 0), A000142, A000670, A226513 (high-order Fubini numbers), A232472, A232473, A232474, A257565, A338873, A338874.

Cf. A338875 (numerators).

F[n_,m_]:=n!Coefficient[Series[x^m/(x^m-Exp[x]+Sum[x^k/k!,{k,0,m}]),{x,0,n}],x,n]; Table[Denominator[F[n-m,m]],{n,0,9},{m,0,n}]//Flatten

tm(n, m) = {my(m = matrix(n, n, i, j, if (i==1, if (j==1, 1/(m + 1)!, if (j==2, 1)), if (j==1, (-1)^(i+1)/(m + i)!)))); for (i=2, n, for (j=2, n, m[i, j] = m[i-1, j-1]; ); ); m; }
T(n, m) = denominator(n!*matdet(tm(n, m))); \\ Michel Marcus, Dec 31 2020

T(n, m) = denominator(F(n, m)).
F(n, m) = n!*det(M(n, m)) where M(n, m) is the n X n Toeplitz matrix whose first row consists in 1/(m + 1)!, 1, 0, ..., 0 and whose first column consists in 1/(m + 1)!, -1/(m + 2)!, ..., (-1)^(n-1)/(m + n)! (see Proposition 5.1 in Komatsu).
F(n, m) = n!*Sum_{k=0..n-1} F(k, m)/((n - k + m)!*k!) for n > 0 and m >= 0 with F(0, m) = 1 (see Lemma 5.2).
F(n, m) = [x^n] n!*x^m/(x^m - exp(x) + E_m(x)), where E_m(x) = Sum_{n=0..m} x^n/n! (see Theorem 5.3 in Komatsu).
F(n, m) = n!*Sum_{k=1..n} Sum_{i_1+...+i_k=n, i_1,...,i_k>=1} Product_{j=1..k} 1/(i_j + m)! for n > 0 and m >= 0 (see Theorem 5.4).
F(1, m) = 1/(m + 1)! (see Theorem 5.5 in Komatsu).
F(n, m) = n!*Sum_{t_1+2*t_2+...+n*t_n=n} (t_1,...,t_n)!*(-1)^(n-t_1-...-t_n)*Product_{j=1..n} (1/(m + j)!)^t_j for n >= m >= 1 (see Theorem 5.7 in Komatsu).
(-1)^(n-1)/(n + m)! = det(M(n, m)) where M(n, m) is the n X n Toeplitz matrix whose first row consists in F(1, m), 1, 0, ..., 0 and whose first column consists in F(1, m), F(2, m)/2!, ..., F(n, m)/n! for n > 0 (see Theorem 5.8 in Komatsu).
Sum_{k=0..n} binomial(n, k)*F(k, m)*F(n-k, m) = - n!/(m^2*m!)*Sum_{l=0..n-1} ((m! + 1)/(m*m!))^(n-l-1)*(l*(m! + 1) - m)/l!*F(l, m) - (n - m)/m*F(n, m) for m > 0 (see Theorem 5.11 in Komatsu).

cp's OEIS Frontend

A003154 Centered 12-gonal numbers, or centered dodecagonal numbers: numbers of the form 6*k*(k-1) + 1.

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A321189 a(n) = n! * [x^n] 1 - 1/(n - 1/(exp(x) - 1)).

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A338875 Array T(n, m) read by ascending antidiagonals: numerators of shifted Fubini numbers F(n, m) where m >= 0.

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A338876 Array T(n, m) read by ascending antidiagonals: denominators of shifted Fubini numbers F(n, m) where m >= 0.

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A321000 Array of generalized Bell numbers, read by antidiagonals upwards.

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A331345 a(n) = (1/n^2) * Sum_{k>=1} k^n * (1 - 1/n)^(k - 1).

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A003154 Centered 12-gonal numbers, or centered dodecagonal numbers: numbers of the form 6k(k-1) + 1.