A057711 -id:A057711 - cp's OEIS frontend

A080929 Sequence associated with a(n) = 2a(n-1) + k(k+2)*a(n-2).

1, 3, 12, 40, 120, 336, 896, 2304, 5760, 14080, 33792, 79872, 186368, 430080, 983040, 2228224, 5013504, 11206656, 24903680, 55050240, 121110528, 265289728, 578813952, 1258291200, 2726297600, 5888802816, 12683575296, 27246198784

Offset: 0

Paul Barry, Feb 26 2003

The third column of number triangle A080928.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Essentially the same as A052482.

Cf. A082140, A082141, A082138, A082139, A080951, A080929, A057711.

Concatenation([1], List([1..30], n-> 2^(n-1)*Binomial(n+2,2))); # G. C. Greubel, Jul 23 2019

[n eq 0 select 1 else (n+1)*(n+2)*2^(n-2): n in [0..30]]; // Vincenzo Librandi, Sep 22 2011

[seq (ceil(binomial(n+2,2)*2^(n-1)),n=0..30)]; # Zerinvary Lajos, Nov 01 2006

CoefficientList[Series[(1-x)(1-2x+4x^2)/(1-2x)^3, {x,0,30}], x] (* Michael De Vlieger, Sep 21 2017 *)
Join[{1}, LinearRecurrence[{6,-12,8}, {3,12,40}, 30]] (* G. C. Greubel, Jul 23 2019 *)

vector(30, n, n--; if(n==0,1, 2^(n-1)*binomial(n+2,2) )) \\ G. C. Greubel, Jul 23 2019

[1]+[2^(n-1)*binomial(n+2,2) for n in (1..30)] # G. C. Greubel, Jul 23 2019

G.f.: (1-x)*(1-2*x+4*x^2)/(1-2*x)^3.
For n>0, a(n) = (n+1)*(n+2)*2^(n-2). - Ralf Stephan, Jan 16 2004
a(n) = Sum_{k=0..n} Sum_{i=0..n} (k+1)*binomial(n-1,i). - Wesley Ivan Hurt, Sep 20 2017
From Amiram Eldar, Jan 07 2022: (Start)
Sum_{n>=0} 1/a(n) = 7 - 8*log(2).
Sum_{n>=0} (-1)^n/a(n) = 24*log(3/2) - 9. (End)

A082139 A transform of binomial(n,5).

Original entry on oeis.org

1, 6, 42, 224, 1008, 4032, 14784, 50688, 164736, 512512, 1537536, 4472832, 12673024, 35094528, 95256576, 254017536, 666796032, 1725825024, 4410441728, 11142168576, 27855421440, 68975329280, 169303080960, 412216197120

Offset: 0

Paul Barry, Apr 06 2003

Sixth row of number array A082137. C(n,5) has e.g.f. (x^5/5!)exp(x). The transform averages the binomial and inverse binomial transforms.

			a(0) = (2^(-1) + 0^0/2)*binomial(5,0) = 2*(1/2) = 1 (use 0^0 = 1).

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (12,-60,160,-240,192,-64).

Cf. A080951, A080952, A082138.

Cf. A082140, A082141, A082138, A080951, A080929, A057711.

[(Ceiling(Binomial(n+5, 5)*2^(n-1))) : n in [0..30]]; // Vincenzo Librandi, Sep 22 2011

[seq (ceil(binomial(n+5,5)*2^(n-1)),n=0..23)]; # Zerinvary Lajos, Nov 01 2006

Drop[With[{nmax = 56}, CoefficientList[Series[x^5*Exp[x]*Cosh[x]/5!, {x, 0, nmax}], x]*Range[0, nmax]!], 5] (* or *) Join[{1}, Table[2^(n-1)* Binomial[n+5,n], {n,1,30}]] (* G. C. Greubel, Feb 05 2018 *)

my(x='x+O('x^30)); Vec(serlaplace(x^5*exp(x)*cosh(x)/5!)) \\ G. C. Greubel, Feb 05 2018

Equals 2 * A080952.
a(n) = (2^(n-1) + 0^n/2)*C(n+5, n).
a(n) = Sum_{j=0..n} C(n+5, j+5)*C(j+5, 5)*(1+(-1)^j)/2.
G.f.: (1 -6*x +30*x^2 -80*x^3 +120*x^4 -96*x^5 +32*x^6)/(1-2*x)^6.
E.g.f.: x^5*exp(x)*cosh(x)/5! (preceded by 5 zeros).
a(n) = ceiling(binomial(n+5,5)*2^(n-1)). - Zerinvary Lajos, Nov 01 2006
From Amiram Eldar, Jan 07 2022: (Start)
Sum_{n>=0} 1/a(n) = 20*log(2) - 38/3.
Sum_{n>=0} (-1)^n/a(n) = 1620*log(3/2) - 656. (End)

A082138 A transform of C(n,3).

Original entry on oeis.org

1, 4, 20, 80, 280, 896, 2688, 7680, 21120, 56320, 146432, 372736, 931840, 2293760, 5570560, 13369344, 31752192, 74711040, 174325760, 403701760, 928514048, 2122317824, 4823449600, 10905190400, 24536678400, 54962159616, 122607894528

Offset: 0

Paul Barry, Apr 06 2003

Fourth row of number array A082137. C(n,3) has e.g.f. (x^3/3!)exp(x). The transform averages the binomial and inverse binomial transforms.

			a(0) = (2^(-1) + 0^0/2)*C(3,0) = 2*(1/2) = 1 (using 0^0=1).

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (8,-24,32,-16).

Cf. A080929, A082137, A082139, A082140, A082141, A080951, A057711.

a:=[4,20,80,280];; for n in [5..30] do a[n]:=8*a[n-1]-24*a[n-2] +32*a[n-3]-16*a[n-4]; od; Concatenation([1], a);

[(Ceiling(Binomial(n+3, 3)*2^(n-1))) : n in [0..30]]; // Vincenzo Librandi, Sep 22 2011

[seq (ceil(binomial(n+3,3)*2^(n-1)),n=0..30)]; # Zerinvary Lajos, Nov 01 2006

Join[{1}, LinearRecurrence[{8,-24,32,-16}, {4,20,80,280}, 30]] (* G. C. Greubel, Jul 23 2019 *)

my(x='x+O('x^30)); Vec((1-4*x+12*x^2-16*x^3 + 8*x^4)/(1-2*x)^4) \\ G. C. Greubel, Jul 23 2019

((1-4*x+12*x^2-16*x^3+8*x^4)/(1-2*x)^4).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jul 23 2019

a(n) = (2^(n-1) + 0^n/2)*C(n+3, n).
a(n) = Sum_{j=0..n} C(n+3, j+3)*C(j+3, 3)*(1 + (-1)^j)/2.
G.f.: (1 - 4*x + 12*x^2 - 16*x^3 + 8*x^4)/(1-2*x)^4.
E.g.f.: (x^3/3!)*exp(x)*cosh(x) (preceded by 3 zeros).
a(n) = ceiling(binomial(n+3,3)*2^(n-1)). - Zerinvary Lajos, Nov 01 2006
From Amiram Eldar, Jan 07 2022: (Start)
Sum_{n>=0} 1/a(n) = 12*log(2) - 7.
Sum_{n>=0} (-1)^n/a(n) = 108*log(3/2) - 43. (End)

A322620 E.g.f.: A(x,y) = (cosh(x)cosh(y) + sinh(x) + sinh(y)) / (1 - sinh(x)sinh(y)), where A(x,y) = Sum_{n>=0} Sum_{k>=0} T(n,k) * x^n*y^k/(n+k)!, as a square table of coefficients T(n,k) read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 6, 6, 1, 1, 16, 30, 16, 1, 1, 40, 140, 140, 40, 1, 1, 96, 615, 1040, 615, 96, 1, 1, 224, 2562, 7000, 7000, 2562, 224, 1, 1, 512, 10220, 43904, 68390, 43904, 10220, 512, 1, 1, 1152, 39384, 260736, 605808, 605808, 260736, 39384, 1152, 1, 1, 2560, 147645, 1482240, 4998210, 7322112, 4998210, 1482240, 147645, 2560, 1, 1, 5632, 541310, 8131200, 39032400, 80735424, 80735424, 39032400, 8131200, 541310, 5632, 1, 1, 12288, 1948650, 43310080, 291662415, 831080448, 1161583500, 831080448, 291662415, 43310080, 1948650, 12288, 1

Offset: 0

Paul D. Hanna, Dec 20 2018

Compare to the addition theorem of Jacobi's elliptic functions: cn(x+y) + i*sn(x+y) = (cn(x) + i*sn(x)*dn(y)) * (cn(y) + i*sn(y)*dn(x)) / (1 - k^2*sn(x)^2*sn(y)^2), where the modulus k is implicit.

See A322190 for another description of the e.g.f. of this sequence.

			E.g.f.: A(x,y) = 1 + (1*x + 1*y) + (1*x^2 + 2*x*y + 1*y^2)/2! + (1*x^3 + 6*x^2*y + 6*x*y^2 + 1*y^3)/3! + (1*x^4 + 16*x^3*y + 30*x^2*y^2 + 16*x*y^3 + 1*y^4)/4! + (1*x^5 + 40*x^4*y + 140*x^3*y^2 + 140*x^2*y^3 + 40*x*y^4 + 1*y^5)/5! + (1*x^6 + 96*x^5*y + 615*x^4*y^2 + 1040*x^3*y^3 + 615*x^2*y^4 + 96*x*y^5 + 1*y^6)/6! + (1*x^7 + 224*x^6*y + 2562*x^5*y^2 + 7000*x^4*y^3 + 7000*x^3*y^4 + 2562*x^2*y^5 + 224*x*y^6 + 1*y^7)/7! + (1*x^8 + 512*x^7*y + 10220*x^6*y^2 + 43904*x^5*y^3 + 68390*x^4*y^4 + 43904*x^3*y^5 + 10220*x^2*y^6 + 512*x*y^7 + 1*y^8)/8! + ...
where A(x,y) = (cosh(x)*cosh(y) + sinh(x) + sinh(y)) / (1 - sinh(x)*sinh(y)).
This square table of coefficients of x^n*y^k/(n+k)! in A(x,y) begins
1, 1, 1, 1, 1, 1, 1, 1, 1, ...;
1, 2, 6, 16, 40, 96, 224, 512, 1152, ...;
1, 6, 30, 140, 615, 2562, 10220, 39384, 147645, ...;
1, 16, 140, 1040, 7000, 43904, 260736, 1482240, 8131200, ...;
1, 40, 615, 7000, 68390, 605808, 4998210, 39032400, 291662415, ...;
1, 96, 2562, 43904, 605808, 7322112, 80735424, 831080448, 8105175936, ...;
1, 224, 10220, 260736, 4998210, 80735424, 1161583500, 15355426944, ...;
1, 512, 39384, 1482240, 39032400, 831080448, 15355426944, 256124504064, ...; ...
This sequence may be written as a triangle, starting as
1,
1, 1,
1, 2, 1,
1, 6, 6, 1;
1, 16, 30, 16, 1;
1, 40, 140, 140, 40, 1;
1, 96, 615, 1040, 615, 96, 1;
1, 224, 2562, 7000, 7000, 2562, 224, 1;
1, 512, 10220, 43904, 68390, 43904, 10220, 512, 1;
1, 1152, 39384, 260736, 605808, 605808, 260736, 39384, 1152, 1;
1, 2560, 147645, 1482240, 4998210, 7322112, 4998210, 1482240, 147645, 2560, 1; ...
RELATED SERIES.
The series expansions for C(x,y) and S(x,y) are given by
C(x,y) = 1 + (1*x^2 + 2*x*y + 1*y^2)/2! + (1*x^4 + 16*x^3*y + 30*x^2*y^2 + 16*x*y^3 + 1*y^4)/4! + (1*x^6 + 96*x^5*y + 615*x^4*y^2 + 1040*x^3*y^3 + 615*x^2*y^4 + 96*x*y^5 + 1*y^6)/6! + (1*x^8 + 512*x^7*y + 10220*x^6*y^2 + 43904*x^5*y^3 + 68390*x^4*y^4 + 43904*x^3*y^5 + 10220*x^2*y^6 + 512*x*y^7 + 1*y^8)/8! + ...
S(x,y) = (1*x + 1*y) + (1*x^3 + 6*x^2*y + 6*x*y^2 + 1*y^3)/3! + (1*x^5 + 40*x^4*y + 140*x^3*y^2 + 140*x^2*y^3 + 40*x*y^4 + 1*y^5)/5! + (1*x^7 + 224*x^6*y + 2562*x^5*y^2 + 7000*x^4*y^3 + 7000*x^3*y^4 + 2562*x^2*y^5 + 224*x*y^6 + 1*y^7)/7! + ...
where A(x,y) = C(x,y) + S(x,y) such that C(x,y)^2 - S(x,y)^2 = 1.
The e.g.f. may be written with coefficients of x^n*y^k/(n!*k!), as follows:
A(x,y) = 1 + (1*x + 1*y) + (1*x^2/2! + 1*x*y + 1*y^2/2!) + (1*x^3/3! + 2*x^2*y/2! + 2*x*y^2/2! + 1*y^3/3!) + (1*x^4/4! + 4*x^3*y/3! + 5*x^2*y^2/(2!*2!) + 4*x*y^3/3! + 1*y^4/4!) + (1*x^5/5! + 8*x^4*y/4! + 14*x^3*y^2/(3!*2!) + 14*x^2*y^3/(2!*3!) + 8*x*y^4/4! + 1*y^5/5!) + (1*x^6/6! + 16*x^5*y/5! + 41*x^4*y^2/(4!*2!) + 52*x^3*y^3/(3!*3!) + 41*x^2*y^4/(2!*4!) + 16*x*y^5/5! + 1*y^6/6!) + (1*x^7/7! + 32*x^6*y/6! + 122*x^5*y^2/(5!*2!) + 200*x^4*y^3/(4!*3!) + 200*x^3*y^4/(3!*4!) + 122*x^2*y^5/(2!*5!) + 32*x*y^6/6! + 1*y^7/7!) + (1*x^8/8! + 64*x^7*y/7! + 365*x^6*y^2/(6!*2!) + 784*x^5*y^3/(5!*3!) + 977*x^4*y^4/(4!*4!) + 784*x^3*y^5/(3!*5!) + 365*x^2*y^6/(2!*6!) + 64*x*y^7/7! + 1*y^8/8!) + ...
these coefficients are described by table A322190.

Paul D. Hanna, Table of n, a(n) for n = 0..1325 terms of this square table read by antidiagonals across rows 0..50.

Cf. A322621 (C(x,y)), A322622 (S(x,y)), A322623 (antidiagonal sums), A322624 (main diagonal), A322625, A057711 (column 1).

Cf. A322190, A245140, A245155, A245166.

nmax = 12;
t[n_, k_] := SeriesCoefficient[(Cosh[x] Cosh[y] + Sinh[x] + Sinh[y])/(1 - Sinh[x] Sinh[y]), {x, 0, n}, {y, 0, k}] (n + k)!;
tt = Table[t[n, k], {n, 0, nmax}, {k, 0, nmax}];
T[n_, k_] := tt[[n+1, k+1]];
Table[T[n-k, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 26 2018 *)

{T(n,k) = my(X=x+x*O(x^n),Y=y+y*O(y^k));
C = cosh(X)*cosh(Y)/(1 - sinh(X)*sinh(Y));
S = (sinh(X) + sinh(Y))/(1 - sinh(X)*sinh(Y));
(n+k)!*polcoeff(polcoeff(C + S,n,x),k,y)}
/* Print as a square table */
for(n=0,10,for(k=0,10,print1( T(n,k),", "));print(""))
/* Print as a triangle */
for(n=0,15,for(k=0,n,print1( T(n-k,k),", "));print(""))

E.g.f.: A(x,y) = (cosh(x) + sinh(x)*cosh(y)) * (cosh(y) + sinh(y)*cosh(x)) / (1 - sinh(x)^2*sinh(y)^2).
E.g.f.: A(x,y) = (cosh(x) + sinh(x)*cosh(y)) / (cosh(y) - sinh(y)*cosh(x)).
E.g.f.: A(x,y) = (cosh(y) + sinh(y)*cosh(x)) / (cosh(x) - sinh(x)*cosh(y)).
E.g.f.: A(x,y) = C(x,y) + S(x,y) such that the following identities hold.
(1) C(x,y)^2 - S(x,y)^2 = 1.
(2a) C(x,y) = cosh(x) * cosh(y) / (1 - sinh(x)*sinh(y)).
(2b) S(x,y) = (sinh(x) + sinh(y)) / (1 - sinh(x)*sinh(y)).
(3a) cosh(x) = C(x,y) * cosh(y) / (1 + sinh(y)*S(x,y)).
(3b) sinh(x) = (S(x,y) - sinh(y)) / (1 + sinh(y)*S(x,y)).
(3c) cosh(y) = C(x,y) * cosh(x) / (1 + sinh(x)*S(x,y)).
(3d) sinh(y) = (S(x,y) - sinh(x)) / (1 + sinh(x)*S(x,y)).
(4a) exp(x) = (C(x,y)*cosh(y) + S(x,y) - sinh(y)) / (1 + sinh(y)*S(x,y)).
(4b) exp(y) = (C(x,y)*cosh(x) + S(x,y) - sinh(x)) / (1 + sinh(x)*S(x,y)).
(5a) exp(x) = (C(x,y) + S(x,y)*cosh(y)) * (cosh(y) - sinh(y)*C(x,y)) / (1 - sinh(y)^2*S(x,y)^2).
(5b) exp(y) = (C(x,y) + S(x,y)*cosh(x)) * (cosh(x) - sinh(x)*C(x,y)) / (1 - sinh(x)^2*S(x,y)^2).
(6a) exp(x) = (C(x,y) + S(x,y)*cosh(y)) / (cosh(y) + sinh(y)*C(x,y)).
(6b) exp(y) = (C(x,y) + S(x,y)*cosh(x)) / (cosh(x) + sinh(x)*C(x,y)).
SPECIAL ARGUMENTS.
A(x, y=0) = exp(x).
A(x, y=x) = (1 + sinh(x)) / (1 - sinh(x)).
A(x, y=-x) = 1.
FORMULAS FOR TERMS.
a(n) = binomial(n,k) * A322190(n,k).
Sum_{k=0..n} 2^k * T(n,k) = A245140(n).
Sum_{k=0..n} 3^k * T(n,k) = A245155(n).
Sum_{k=0..n} 2^(n-k) * 3^k * T(n,k) = A245166(n).

A078836 a(n) = n*2^(n-6).

Original entry on oeis.org

6, 14, 32, 72, 160, 352, 768, 1664, 3584, 7680, 16384, 34816, 73728, 155648, 327680, 688128, 1441792, 3014656, 6291456, 13107200, 27262976, 56623104, 117440512, 243269632, 503316480, 1040187392, 2147483648, 4429185024, 9126805504, 18790481920, 38654705664

Offset: 6

Silvia Heubach (sheubac(AT)calstatela.edu), Jan 17 2003

a(n) is the number of occurrences of 5s in the palindromic compositions of 2n-1 = the number of occurrences of 6s in the palindromic compositions of 2n.
This sequence is part of a family of sequences, namely R(n,k), the number of ks in palindromic compositions of n. See also A057711, A001792, A079859, A079861 - A079863. General formula: R(n,k)=2^(floor(n/2) - k) * (2 + floor(n/2) - k) if n and k have different parity and R(n,k)=2^(floor(n/2) - k) * (2 + floor(n/2) - k + 2^(floor((k+1)/2 - 1)) otherwise, for n >= 2k.
Also the number of independent vertex sets and vertex covers in the (n-4)-sun graph. - Eric W. Weisstein, Sep 27 2017

			a(6) = 6 since the palindromic compositions of 11 that contain a 5 are 3+5+3, 1+2+5+2+1, 2+1+5+1+2, 1+1+1+5+1+1+1 and 5+1+5, for a total of 6 5s. The palindromic compositions of 12 that contain a 6 are 3+6+3, 1+2+6+2+1, 2+1+6+1+2, 1+1+1+6+1+1+1 and 6+6.

Vincenzo Librandi, Table of n, a(n) for n = 6..3000
Phyllis Chinn, Ralph Grimaldi and Silvia Heubach, The frequency of summands of a particular size in Palindromic Compositions, Ars Combin., Vol. 69 (2003), pp. 65-78.
Eric Weisstein's World of Mathematics, Independent Vertex Set.
Eric Weisstein's World of Mathematics, Sun Graph.
Eric Weisstein's World of Mathematics, Vertex Cover.
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A057711, A001792, A079859, A079861, A079862, A079863.

[n*2^(n-6): n in [6..40]]; // Vincenzo Librandi, Oct 04 2011

Table[n 2^(n - 6), {m, 6, 50}]
LinearRecurrence[{4, -4}, {6, 14}, 20] (* Eric W. Weisstein, Sep 27 2017 *)
CoefficientList[Series[-2 (-3 + 5 x)/(-1 + 2 x)^2, {x, 0, 20}], x] (* Eric W. Weisstein, Sep 27 2017 *)

a(n)=n<<(n-6) \\ Charles R Greathouse IV, Oct 03 2011

Vec(-2*x^6*(5*x-3)/(2*x-1)^2 + O(x^100)) \\ Colin Barker, Sep 29 2015

def a(n): return n << (n-6)
print([a(n) for n in range(6, 37)]) # Michael S. Branicky, Jun 14 2021

From Colin Barker, Sep 29 2015: (Start)
a(n) = 2*A045891(n-4).
a(n) = 4*a(n-1) - 4*a(n-2) for n > 7.
G.f.: -2*x^6*(5*x-3) / (2*x-1)^2.
(End)
From Amiram Eldar, Jan 12 2021: (Start)
Sum_{n>=6} 1/a(n) = 64*log(2) - 661/15.
Sum_{n>=6} (-1)^n/a(n) = 391/15 - 64*log(3/2). (End)

A080951 Sequence associated with recurrence a(n) = 2a(n-1) + k(k+2)*a(n-2).

Original entry on oeis.org

1, 5, 30, 140, 560, 2016, 6720, 21120, 63360, 183040, 512512, 1397760, 3727360, 9748480, 25067520, 63504384, 158760960, 392232960, 958791680, 2321285120, 5571084288, 13264486400, 31352422400, 73610035200, 171756748800, 398475657216, 919559208960, 2111580405760

Offset: 0

Paul Barry, Feb 26 2003

Fifth column of triangle A080928.

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (10,-40,80,-80,32).

Cf. A082140, A082141, A082138, A082139, A080928, A080929, A057711.

a:=[5,30,140,560,2016];; for n in [6..30] do a[n]:=10*a[n-1] -40*a[n-2]+80*a[n-3]-80*a[n-4]+32*a[n-5]; od; Concatenation([1], a); # G. C. Greubel, Jul 23 2019

[(Ceiling(Binomial(n+4, 4)*2^(n-1))) : n in [0..30]]; // Vincenzo Librandi, Sep 22 2011

[seq( ceil(binomial(n+4,4)*2^(n-1)),n=0..30)]; # Zerinvary Lajos, Nov 01 2006

Join[{1}, LinearRecurrence[{10,-40,80,-80,32}, {5,30,140,560,2016}, 30]] (* G. C. Greubel, Jul 23 2019 *)

my(x='x+O('x^30)); Vec((1-x)*(1-4*x+16*x^2-24*x^3 +16*x^4)/(1 -2*x)^5) \\ G. C. Greubel, Jul 23 2019

((1-x)*(1-4*x+16*x^2-24*x^3+16*x^4)/(1-2*x)^5).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jul 23 2019

G.f.: (1-x)*(1 - 4*x + 16*x^2 - 24*x^3 + 16*x^4)/(1-2*x)^5.
a(n) = ceiling(binomial(n+4,4)*2^(n-1)). - Zerinvary Lajos, Nov 01 2006
From Amiram Eldar, Jan 07 2022: (Start)
Sum_{n>=0} 1/a(n) = 37/3 - 16*log(2).
Sum_{n>=0} (-1)^n/a(n) = 432*log(3/2) - 523/3. (End)
E.g.f.: (3 + exp(2*x)*(3 + 24*x + 36*x^2 + 16*x^3 + 2*x^4))/6. - Stefano Spezia, Sep 02 2025

A082134 Expansion of e.g.f. xexp(3x)*cosh(x).

Original entry on oeis.org

0, 1, 6, 30, 144, 680, 3168, 14560, 66048, 296064, 1313280, 5772800, 25178112, 109078528, 469819392, 2013388800, 8590196736, 36507779072, 154620002304, 652837519360, 2748784312320, 11544883101696, 48378534690816

Offset: 0

Paul Barry, Apr 06 2003

Binomial transform of A082133. 3rd binomial transform of (0,1,0,3,0,5,0,7,...)
Let P(A) be the power set of an n-element set A and B be the Cartesian product of P(A) with itself.  Then remove (y,x) from B when (x,y) is in B and x <> y and call this R35.  Then a(n) = the sum of the size of the intersection of x and y for every (x,y) of R35. - Ross La Haye, Dec 30 2007; edited Jan 05 2013
A133224 is the analogous sequence if "Intersection" is replaced by "Union" and A002697 is the analogous sequence if "Intersection" is replaced by "Symmetric difference". Here, X Intersection Y = Y Intersection X is considered as the same set [Relation (37): T_Q(n) in document of Ross La Haye in reference]. If we want to consider that X Intersection Y and Y Intersection X are two distinct formula for describing the same set, see A002697. - Bernard Schott, Jan 19 2013

G. C. Greubel, Table of n, a(n) for n = 0..1000
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
Index entries for linear recurrences with constant coefficients, signature (12,-52,96,-64).

Cf. A057711 (x*exp(x)*cosh(x)), A082133 (x*exp(2*x)*cosh(x)).
Cf. A082135 (x*exp(4*x)*cosh(x)), A082136 (x*exp(5*x)*cosh(x)).
Cf. A002697, A133224.

[n*2^(n-2)*(1+2^(n-1)): n in [0..25]]; // G. C. Greubel, Feb 05 2018

a:= n -> n*binomial(2^(n-1) +1, 2); seq(a(n), n=0..25); # G. C. Greubel, Apr 16 2020

Table[n(2^(n-1) +4^(n-1))/2, {n, 0, 22}] (* Michael De Vlieger, Nov 29 2015 *)
With[{nmax = 25}, CoefficientList[Series[x*Exp[3*x]*Cosh[x], {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Feb 05 2018 *)

a(n)=n*(2^n--+4^n)/2 \\ Charles R Greathouse IV, Jan 14 2013

[n*binomial(2^(n-1)+1, 2) for n in (0..25)] # G. C. Greubel, Apr 16 2020

a(n) = n*(2^(n-1) + 4^(n-1))/2.
E.g.f.: x*exp(3*x)*cosh(x).
Conjecture: (n+28)*a(n) + (n-282)*a(n-1) + 2*(-17*n+423)*a(n-2) + 8*(7*n-94)*a(n-3) = 0. - R. J. Mathar, Nov 29 2012
G.f.: x*(10*x^2-6*x+1) / ((2*x-1)^2*(4*x-1)^2). - Colin Barker, Dec 10 2012

A099035 a(n) = (n+1)*2^(n-1) - 1.

Original entry on oeis.org

1, 5, 15, 39, 95, 223, 511, 1151, 2559, 5631, 12287, 26623, 57343, 122879, 262143, 557055, 1179647, 2490367, 5242879, 11010047, 23068671, 48234495, 100663295, 209715199, 436207615, 905969663, 1879048191, 3892314111, 8053063679

Offset: 1

Ralf Stephan, Sep 28 2004

Row sums of triangle A135852. - Gary W. Adamson, Dec 01 2007

Binomial transform of [1, 4, 6, 8, 10, 12, 14, 16, ...]. Equals A128064 * A000225, (A000225 starting 1, 3, 7, 15, ...). - Gary W. Adamson, Dec 28 2007

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-8,4).

First differences of A066524.

Cf. A135852, A128064, A000225.

[(n+1)*2^(n-1) -1: n in [1..30]]; // G. C. Greubel, Dec 31 2017

Table[(n + 1)*2^(n - 1) - 1, {n,1,30}] (* G. C. Greubel, Dec 31 2017 *)
LinearRecurrence[{5,-8,4},{1,5,15},30] (* Harvey P. Dale, Dec 28 2022 *)

a(n)=(n+1)*2^(n-1)-1 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = A057711(n+1) - 1 = A058966(n+3)/2 = (A087323(n)-1)/2 = (A074494(n+1)-2)/3 = (A003261(n+1)-3)/4 = A036289(n+1)/4 - 1, n>0.
a(n) = A131056(n+1) - 2. - Juri-Stepan Gerasimov, Oct 02 2011
From Colin Barker, Mar 23 2012: (Start)
a(n) = 5*a(n-1) - 8*a(n-2) + 4*a(n-3).
G.f.: x*(1-2*x^2)/((1-x)*(1-2*x)^2). (End)
E.g.f.: ((2*x+1)*exp(2*x) - 2*exp(x) + 1)/2. - G. C. Greubel, Dec 31 2017

A079859 a(n) = n*2^(n-4).

Original entry on oeis.org

4, 10, 24, 56, 128, 288, 640, 1408, 3072, 6656, 14336, 30720, 65536, 139264, 294912, 622592, 1310720, 2752512, 5767168, 12058624, 25165824, 52428800, 109051904, 226492416, 469762048, 973078528, 2013265920, 4160749568, 8589934592, 17716740096, 36507222016

Offset: 4

Silvia Heubach (sheubac(AT)calstatela.edu), Jan 11 2003

a(n) = the number of occurrences of 3s in the palindromic compositions of m = 2*n-1 = the number of occurrences of 4s in the palindromic compositions of k = 2*n.
This sequence is part of a family of sequences, namely R(n,k), the number of ks in palindromic compositions of n. See also A057711, A001792, A078836, A079861, A079862, A079863. General formula: R(n,k)=2^(floor(n/2) - k) * (2 + floor(n/2) - k) if n and k have different parity and R(n,k)=2^(floor(n/2) - k) * (2 + floor(n/2) - k + 2^(floor((k+1)/2 - 1)) otherwise, for n >= 2k.
Number of 2 X n binary matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (10;0) and (01;1). An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1Sergey Kitaev, Nov 11 2004
a(n) appears to be the coefficient of Pi^n in the closed-form expression for the expected value of X^n, where X is the area of a spherical triangle formed by three random points on the unit sphere. (The n*2^(n-4) formula applies when n=2,3 as well, and produces fractional coefficients.) - Drake Thomas, Jan 24 2021

			a(4)=4 since the palindromic compositions of 7 that contain a 3 are 2+3+2, 1+1+3+1+1 and 3+1+3, for a total of 4 3s. The palindromic compositions of 8 that contain a 4 are 2+4+2, 1+1+4+1+1 and 4+4.

Vincenzo Librandi, Table of n, a(n) for n = 4..1000
Phyllis Chinn, Ralph Grimaldi and Silvia Heubach, The frequency of summands of a particular size in Palindromic Compositions, Ars Combin., Vol. 69 (2003), pp. 65-78.
Sergey Kitaev, On multi-avoidance of right angled numbered polyomino patterns, Integers: Electronic Journal of Combinatorial Number Theory, Vol. 4 (2004), A21, 20pp.
Sergey Kitaev, On multi-avoidance of right angled numbered polyomino patterns, University of Kentucky Research Reports (2004).
Math StackExchange, The distribution of areas of a random triangle on the sphere.
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A057711, A001792, A078836, A079861, A079862.

Main diagonal of A049089.

[n*2^(n-4) : n in [4..40]]; // Vincenzo Librandi, Sep 22 2011

Mathematica
```
Table[i*2^(i - 4), {i, 4, 50}]
```

Vec(-2*x^4*(3*x-2)/(2*x-1)^2 + O(x^50)) \\ Colin Barker, Sep 29 2015

a(n) = n*2^(n-4);
vector(40, n, a(n+3)) \\ Altug Alkan, Sep 29 2015

O.g.f.: 2*x^4*(2-3*x)/(1-2*x)^2. a(n) = 2*A045623(n-3). - R. J. Mathar, Jun 13 2008
a(n) = 4*a(n-1) - 4*a(n-2) for n>5. - Colin Barker, Sep 29 2015
From Amiram Eldar, Jan 12 2021: (Start)
Sum_{n>=4} 1/a(n) = 16*log(2) - 32/3.
Sum_{n>=4} (-1)^n/a(n) = 20/3 - 16*log(3/2). (End)
E.g.f.: x*(exp(2*x) - 1 - 2*x - 2*x^2)/8. - Stefano Spezia, Apr 06 2021

A079861 a(n) is the number of occurrences of 7's in the palindromic compositions of 2n-1, or also, the number of occurrences of 8's in the palindromic compositions of 2n.

Original entry on oeis.org

10, 22, 48, 104, 224, 480, 1024, 2176, 4608, 9728, 20480, 43008, 90112, 188416, 393216, 819200, 1703936, 3538944, 7340032, 15204352, 31457280, 65011712, 134217728, 276824064, 570425344, 1174405120, 2415919104, 4966055936

Offset: 8

Silvia Heubach (sheubac(AT)calstatela.edu), Jan 11 2003

This sequence is part of a family of sequences, namely R(n,k), the number of k's in palindromic compositions of n. See also A057711, A001792, A078836, A079861, A079862. General formula: R(n,k) = 2^(floor(n/2) - k) * (2 + floor(n/2) - k) if n and k have different parity and R(n,k) = 2^(floor(n/2) - k) * (2 + floor(n/2) - k + 2^(floor((k+1)/2 - 1)) otherwise, for n >= 2*k.

			a(8)=10 since the palindromic compositions of 15 that contain a 7 are 7+1+7, 4+7+4, 1+3+7+3+1, 3+1+7+1+3, 2+2+7+2+2, 1+1+1+1+7+1+1+1+1, 1+1+2+7+2+1+1, 1+2+1+7+1+2+1 and 2+1+1+7+1+1+2, for a total of 10 7's.

Vincenzo Librandi, Table of n, a(n) for n = 8..1000
Phyllis Chinn, Ralph Grimaldi and Silvia Heubach, The Frequency of Summands of a Particular Size in Palindromic Compositions, Ars Combin., Vol. 69 (2003), pp. 65-78.
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A057711, A001792, A079859, A061256, A079862, A079863, A111297.

[(2+n)*2^(n-8) : n in [8..40]]; // Vincenzo Librandi, Sep 22 2011

Table[(2 + i)*2^(i - 8), {i, 8, 50}]
LinearRecurrence[{4,-4},{10,22},50] (* Harvey P. Dale, Jun 04 2025 *)

Vec(-2*x^8*(9*x-5)/(2*x-1)^2 + O(x^100)) \\ Colin Barker, Dec 16 2014

a(n) = (2+n)*2^(n-8).
a(n) = 2*A111297(n-6). - Colin Barker, Dec 16 2014
a(n) = 4*a(n-1) - 4*a(n-2). - Colin Barker, Dec 16 2014
G.f.: -2*x^8*(9*x-5) / (2*x-1)^2. - Colin Barker, Dec 16 2014
From Amiram Eldar, Jan 13 2021: (Start)
Sum_{n>=8} 1/a(n) = 1024*log(2) - 447047/630.
Sum_{n>=8} (-1)^n/a(n) = 261617/630 - 1024*log(3/2). (End)

cp's OEIS Frontend

A080929 Sequence associated with a(n) = 2*a(n-1) + k*(k+2)*a(n-2).

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A082139 A transform of binomial(n,5).

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A082138 A transform of C(n,3).

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A322620 E.g.f.: A(x,y) = (cosh(x)*cosh(y) + sinh(x) + sinh(y)) / (1 - sinh(x)*sinh(y)), where A(x,y) = Sum_{n>=0} Sum_{k>=0} T(n,k) * x^n*y^k/(n+k)!, as a square table of coefficients T(n,k) read by antidiagonals.

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A078836 a(n) = n*2^(n-6).

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A080951 Sequence associated with recurrence a(n) = 2*a(n-1) + k*(k+2)*a(n-2).

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A080929 Sequence associated with a(n) = 2a(n-1) + k(k+2)*a(n-2).

A322620 E.g.f.: A(x,y) = (cosh(x)cosh(y) + sinh(x) + sinh(y)) / (1 - sinh(x)sinh(y)), where A(x,y) = Sum_{n>=0} Sum_{k>=0} T(n,k) * x^n*y^k/(n+k)!, as a square table of coefficients T(n,k) read by antidiagonals.

A080951 Sequence associated with recurrence a(n) = 2a(n-1) + k(k+2)*a(n-2).

A082134 Expansion of e.g.f. xexp(3x)*cosh(x).

A079861 a(n) is the number of occurrences of 7's in the palindromic compositions of 2n-1, or also, the number of occurrences of 8's in the palindromic compositions of 2n.