A053132 -id:A053132 - cp's OEIS frontend

A005995 Alkane (or paraffin) numbers l(8,n).

1, 3, 12, 28, 66, 126, 236, 396, 651, 1001, 1512, 2184, 3108, 4284, 5832, 7752, 10197, 13167, 16852, 21252, 26598, 32890, 40404, 49140, 59423, 71253, 85008, 100688, 118728, 139128, 162384, 188496, 218025, 250971, 287964, 329004, 374794, 425334, 481404, 543004

Offset: 0

N. J. A. Sloane, Winston C. Yang (yang(AT)math.wisc.edu)

From M. F. Hasler, May 01 2009: (Start)
Also, number of 5-element subsets of {1,...,n+5} whose elements sum to an odd integer, i.e. column 5 of A159916.
A linear recurrent sequence with constant coefficients and characteristic polynomial x^9 - 3*x^8 + 8*x^6 - 6*x^5 - 6*x^4 + 8*x^3 - 3*x + 1. (End)
Equals (1/2)*((1, 6, 21, 56, 126, 252, ...) + (1, 0, 3, 0, 6, 0, 10, ...)), see A000389 and A000217.
Equals row sums of triangle A160770.
F(1,5,n) is the number of bracelets with 1 blue, 5 identical red and n identical black beads. If F(1,5,1) = 3 and F(1,5,2) = 12 taken as a base, F(1,5,n) = n(n+1)(n+2)(n+3)/24 + F(1,3,n) + F(1,5,n-2). [F(1,3,n) is the number of bracelets with 1 blue, 3 identical red and n identical black beads. If F(1,3,1) = 2 and F(1,3,2) = 6 taken as a base F(1,3,n) = n(n+1)/2 + [|n/2|] + 1 + F(1,3,n-2)], where [|x|]: if a is an integer and a<=x

S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Winston C. Yang (paper in preparation).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Johann Cigler, Some remarks on Rogers-Szegö polynomials and Losanitsch's triangle, arXiv:1711.03340 [math.CO], 2017.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)
N. J. A. Sloane, Classic Sequences
L. Smith, Polynomial invariants of finite groups. A survey of recent developments. Bull. Amer. Math. Soc. (N.S.) 34 (1997), no. 3, 211-250.
Ata A. Uslu and Hamdi G. Ozmenekse, F(1,3,n)
Ata A. Uslu and Hamdi G. Ozmenekse, F(1,5,n)
Index entries for linear recurrences with constant coefficients, signature (3, 0, -8, 6, 6, -8, 0, 3, -1).

Cf. A160770, A053132 (bisection), A271870 (bisection), A018210 (partial sums).

a:= n-> (Matrix([[1, 0$6, -3, -9]]). Matrix(9, (i,j)-> if (i=j-1) then 1 elif j=1 then [3, 0, -8, 6, 6, -8, 0, 3, -1][i] else 0 fi)^n)[1, 1]: seq(a(n), n=0..40); # Alois P. Heinz, Jul 31 2008

a[n_?OddQ] := 1/240*(n+1)*(n+2)*(n+3)*(n+4)*(n+5); a[n_?EvenQ] := 1/240*(n+2)*(n+4)*(n+6)*(n^2+3*n+5); Table[a[n], {n, 0, 32}] (* Jean-François Alcover, Mar 17 2014, after M. F. Hasler *)
LinearRecurrence[{3, 0, -8, 6, 6, -8, 0, 3, -1},{1, 3, 12, 28, 66, 126, 236, 396, 651},40] (* Ray Chandler, Sep 23 2015 *)

a(k)= if(k%2,(k+1)*(k+3)*(k+5),(k+6)*(k^2+3*k+5))*(k+2)*(k+4)/240 \\ M. F. Hasler, May 01 2009

G.f.: (1+3*x^2)/((1-x)^3*(1-x^2)^3).
a(2n-1) = n(n+1)(n+2)(2n+1)(2n+3)/30, a(2n) = (n+1)(n+2)(n+3)(4n^2+6n+5)/ 30. [M. F. Hasler, May 01 2009]
a(n) = (1/240)*(n+1)*(n+2)*(n+3)*(n+4)*(n+5) + (1/16)*(n+2)*(n+4)*(1/2)*(1+(-1)^n). [Yosu Yurramendi, Jun 20 2013]
a(n) = A038163(n)+3*A038163(n-2). - R. J. Mathar, Mar 29 2018

A053127 Binomial coefficients C(2*n-4,5).

Original entry on oeis.org

6, 56, 252, 792, 2002, 4368, 8568, 15504, 26334, 42504, 65780, 98280, 142506, 201376, 278256, 376992, 501942, 658008, 850668, 1086008, 1370754, 1712304, 2118760, 2598960, 3162510, 3819816, 4582116, 5461512, 6471002, 7624512

Offset: 5

Wolfdieter Lang

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings).

Vincenzo Librandi, Table of n, a(n) for n = 5..200
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].
Milan Janjic, Two Enumerative Functions, University of Banja Luka (Bosnia and Herzegovina, 2017).
Ângela Mestre and José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A053123, A053132, A053126.

a053127 = (* 2) . a053132  -- Reinhard Zumkeller, Mar 03 2015

[Binomial(2*n-4,5): n in [5..40]]; // Vincenzo Librandi, Oct 07 2011

Binomial[2Range[5,40]-4,5] (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{6,56,252,792,2002,4368},30] (* Harvey P. Dale, Jun 03 2013 *)

for(n=5,50, print1(binomial(2*n-4,5), ", ")) \\ G. C. Greubel, Aug 26 2018

a(n) = binomial(2*n-4, 5) if n >= 5 else 0.
a(n) = -A053123(n,5), n >= 5; a(n) := 0, n=0..4 (sixth column of shifted Chebyshev's S-triangle, decreasing order).
G.f.: (6+20*x+6*x^2)/(1-x)^6.
a(n) = 6*a(n-1) - 15*a(n-2) +20*a(n-3) -15*a(n-4) +6*a(n-5) -a(n-6). - Harvey P. Dale, Jun 03 2013
E.g.f.: (-840 + 750*x - 330*x^2 + 95*x^3 - 20*x^4 + 4*x^5)*exp(x)/15. - G. C. Greubel, Aug 26 2018
a(n) = (2*n-8)*(2*n-7)*(2*n-6)*(2*n-5)*(2*n-4)/120. - Wesley Ivan Hurt, Mar 25 2020
From Amiram Eldar, Jan 03 2022: (Start)
Sum_{n>=5} 1/a(n) = 335/12 - 40*log(2).
Sum_{n>=5} (-1)^(n+1)/a(n) = 85/12 - 10*log(2). (End)

A060082 Coefficients of even-indexed Euler polynomials (falling powers without zeros).

Original entry on oeis.org

1, 1, -1, 1, -2, 1, 1, -3, 5, -3, 1, -4, 14, -28, 17, 1, -5, 30, -126, 255, -155, 1, -6, 55, -396, 1683, -3410, 2073, 1, -7, 91, -1001, 7293, -31031, 62881, -38227, 1, -8, 140, -2184, 24310, -177320, 754572, -1529080, 929569, 1, -9, 204, -4284, 67626, -753610, 5497596, -23394924, 47408019

Offset: 0

Wolfdieter Lang, Mar 29 2001

E(2n,x) = x^(2n) + Sum_{k=1..n} a(n,k)*x^(2n-2k+1).

			E(0,x) = 1.
E(2,x) = x^2 - x.
E(4,x) = x^4 - 2*x^3 + x.
E(6,x) = x^6 - 3*x^5 + 5*x^3 - 3*x.
E(8,x) = x^8 - 4*x^7 + 14*x^5 - 28*x^3 + 17*x.
E(10,x) = x^10 - 5*x^9 + 30*x^7 - 126*x^5 + 255*x^3 - 155*x.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 809.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
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].
Z.-W. Sun, Introduction to Bernoulli and Euler polynomials

E(2n, 1/2)*(-4)^n = A000364(n) (signless Euler numbers without zeros).
-E(2n, -1/2)*(-4)^n/3 = A076552(n), -E(2n, 1/3)*(-9)^n/2 = A002114(n).
Cf. A060083 (rising powers), A060096-7 (Euler polynomials), A004172 (with zeros).
Columns (left edge) include A000330, A053132. Columns (right edge) include A001469.

Table[ CoefficientList[ EulerE[2*n, x], x] // Reverse // DeleteCases[#, 0]&, {n, 0, 9}] // Flatten (* Jean-François Alcover, Jun 21 2013 *)

{B(n,v='x)=sum(i=0,n,binomial(n,i)*bernfrac(i)*v^(n-i))} E(n,v='x)=2/(n+1)*(B(n+1,v)-2^(n+1)*B(n+1,v/2)) \\ Ralf Stephan, Nov 05 2004

E(n, x) = 2/(n+1) * [B(n+1, x) - 2^(n+1)*B(n+1, x/2) ], with B(n, x) the Bernoulli polynomials.

Edited by Ralf Stephan, Nov 05 2004

A091036 Sixth column (k=7) of array A090438 ((4,2)-Stirling2) divided by 48=4!*2.

Original entry on oeis.org

1, 840, 498960, 285405120, 173145772800, 115598414131200, 86165279456256000, 72034173625430016000, 67538393730337001472000, 70856069211827240140800000, 82901600977837870964736000000

Offset: 4

Wolfdieter Lang, Jan 23 2004

Cf. A091035 (fifth column of A090438).

A091036 := proc(n)
    binomial(2*n-2,5)*(2*n)!/7!/4!/2 ;
end proc:
seq(A091036(n),n=4..40) ; # R. J. Mathar, Jul 27 2022

a(n)=A090438(n, 7)/48, n>=4.
a(n)=binomial(2*n-2, 5)*(2*n)!/(7!*4!*2)= A053132(n+1)*(2*n)!/(7!*4!), n>=4.
E.g.f.:(sum(((-1)^(p+1))*binomial(7, p)*hypergeom([(p-1)/2, p/2], [], 4*x), p=2..7) + 6)/(7!*48) (cf. A090438).
D-finite with recurrence (2*n-7)*(n-4)*a(n) -2*n*(n-1)*(2*n-1)*(2*n-3)*a(n-1)=0. - R. J. Mathar, Jul 27 2022

A196790 Binomial coefficients C(2*n-9,10).

Original entry on oeis.org

11, 286, 3003, 19448, 92378, 352716, 1144066, 3268760, 8436285, 20030010, 44352165, 92561040, 183579396, 348330136, 635745396, 1121099408, 1917334783, 3190187286, 5178066751, 8217822536, 12777711870, 19499099620, 29248649430, 43183019880, 62828356305, 90177170226

Offset: 10

Vincenzo Librandi, Oct 07 2011

Vincenzo Librandi, Table of n, a(n) for n = 10..200

Cf. A053126, A053127, A053128, A053129, A053130, A053131, A053132, A053133.

Magma
```
[Binomial(2*n-9,10): n in [10..40]];
```

a[n_] := Binomial[2*n - 9, 10]; Array[a, 20, 10] (* Amiram Eldar, Oct 21 2022 *)

G.f.: x^10*(11+165*x+462*x^2+330*x^3+55*x^4+x^5) / (1-x)^11. - R. J. Mathar, Oct 08 2011
From Amiram Eldar, Oct 21 2022: (Start)
Sum_{n>=10} 1/a(n) = 447187/252 - 2560*log(2).
Sum_{n>=10} (-1)^n/a(n) = 40*Pi + 80*log(2) - 6517/36. (End)

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A005995 Alkane (or paraffin) numbers l(8,n).

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A053127 Binomial coefficients C(2*n-4,5).

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A060082 Coefficients of even-indexed Euler polynomials (falling powers without zeros).

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A091036 Sixth column (k=7) of array A090438 ((4,2)-Stirling2) divided by 48=4!*2.

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A196790 Binomial coefficients C(2*n-9,10).

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