A085528 -id:A085528 - cp's OEIS frontend

A374848 Obverse convolution A000045**A000045; see Comments.

0, 1, 2, 16, 162, 3600, 147456, 12320100, 2058386904, 701841817600, 488286500625000, 696425232679321600, 2038348954317776486400, 12259459134020160144810000, 151596002479762016373851690400, 3855806813438155578522841251840000

Offset: 0

Clark Kimberling, Jul 31 2024

nonn

The obverse convolution of sequences
  s = (s(0), s(1), ...) and t = (t(0), t(1), ...)
is introduced here as the sequence s**t given by
s**t(n) = (s(0)+t(n)) * (s(1)+t(n-1)) * ... * (s(n)+t(0)).
Swapping * and + in the representation s(0)*t(n) + s(1)*t(n-1) + ... + s(n)*t(0)
of ordinary convolution yields s**t.
If x is an indeterminate or real (or complex) variable, then for every sequence t of real (or complex) numbers, s**t is a sequence of polynomials p(n) in x, and the zeros of p(n) are the numbers -t(0), -t(1), ..., -t(n).
Following are abbreviations in the guide below for triples (s, t, s**t):
F = (0,1,1,2,3,5,...) = A000045, Fibonacci numbers
L = (2,1,3,4,7,11,...) = A000032, Lucas numbers
P = (2,3,5,7,11,...) = A000040, primes
T = (1,3,6,10,15,...) = A000217, triangular numbers
C = (1,2,6,20,70, ...) = A000984, central binomial coefficients
LW = (1,3,4,6,8,9,...) = A000201, lower Wythoff sequence
UW = (2,5,7,10,13,...) = A001950, upper Wythoff sequence
[ ] = floor
In the guide below, sequences s**t are identified with index numbers Axxxxxx; in some cases, s**t and Axxxxxx differ in one or two initial terms.
Table 1. s = A000012 = (1,1,1,1...) = (1);
t = A000012; 1                 s**t = A000079; 2^(n+1)
t = A000027; n                 s**t = A000142; (n+1)!
t = A000040, P                 s**t = A054640
t = A000040, P           (1/3) s**t = A374852
t = A000079, 2^n               s**t = A028361
t = A000079, 2^n         (1/3) s**t = A028362
t = A000045, F                 s**t = A082480
t = A000032, L                 s**t = A374890
t = A000201, LW                s**t = A374860
t = A001950, UW                s**t = A374864
t = A005408, 2*n+1             s**t = A000165, 2^n*n!
t = A016777, 3*n+1             s**t = A008544
t = A016789, 3*n+2             s**t = A032031
t = A000142, n!                s**t = A217757
t = A000051, 2^n+1             s**t = A139486
t = A000225, 2^n-1             s**t = A006125
t = A032766, [3*n/2]           s**t = A111394
t = A034472, 3^n+1             s**t = A153280
t = A024023, 3^n-1             s**t = A047656
t = A000217, T                 s**t = A128814
t = A000984, C                 s**t = A374891
t = A279019, n^2-n             s**t = A130032
t = A004526, 1+[n/2]           s**t = A010551
t = A002264, 1+[n/3]           s**t = A264557
t = A002265, 1+[n/4]           s**t = A264635
Sequences (c)**L, for c=2..4: A374656 to A374661
Sequences (c)**F, for c=2..6: A374662, A374662, A374982 to A374855
The obverse convolutions listed in Table 1 are, trivially, divisibility sequences. Likewise, if s = (-1,-1,-1,...) instead of s = (1,1,1,...), then s**t is a divisibility sequence for every choice of t; e.g. if s = (-1,-1,-1,...) and t = A279019, then s**t = A130031.
Table 2. s = A000027 = (0,1,2,3,4,5,...) = (n);
t = A000027, n                 s**t = A007778, n^(n+1)
t = A000290, n^2               s**t = A374881
t = A000040, P                 s**t = A374853
t = A000045, F                 s**t = A374857
t = A000032, L                 s**t = A374858
t = A000079, 2^n               s**t = A374859
t = A000201, LW                s**t = A374861
t = A005408, 2*n+1             s**t = A000407, (2*n+1)! / n!
t = A016777, 3*n+1             s**t = A113551
t = A016789, 3*n+2             s**t = A374866
t = A000142, n!                s**t = A374871
t = A032766, [3*n/2]           s**t = A374879
t = A000217, T                 s**t = A374892
t = A000984, C                 s**t = A374893
t = A038608, n*(-1)^n          s**t = A374894
Table 3. s = A000290 = (0,1,4,9,16,...) = (n^2);
t = A000290, n^2               s**t = A323540
t = A002522, n^2+1             s**t = A374884
t = A000217, T                 s**t = A374885
t = A000578, n^3               s**t = A374886
t = A000079, 2^n               s**t = A374887
t = A000225, 2^n-1             s**t = A374888
t = A005408, 2*n+1             s**t = A374889
t = A000045, F                 s**t = A374890
Table 4. s = t;
s = t = A000012, 1             s**s = A000079; 2^(n+1)
s = t = A000027, n             s**s = A007778, n^(n+1)
s = t = A000290, n^2           s**s = A323540
s = t = A000045, F             s**s = this sequence
s = t = A000032, L             s**s = A374850
s = t = A000079, 2^n           s**s = A369673
s = t = A000244, 3^n           s**s = A369674
s = t = A000040, P             s**s = A374851
s = t = A000201, LW            s**s = A374862
s = t = A005408, 2*n+1         s**s = A062971
s = t = A016777, 3*n+1         s**s = A374877
s = t = A016789, 3*n+2         s**s = A374878
s = t = A032766, [3*n/2]       s**s = A374880
s = t = A000217, T             s**s = A375050
s = t = A005563, n^2-1         s**s = A375051
s = t = A279019, n^2-n         s**s = A375056
s = t = A002398, n^2+n         s**s = A375058
s = t = A002061, n^2+n+1       s**s = A375059
If n = 2*k+1, then s**s(n) is a square; specifically,
s**s(n) = ((s(0)+s(n))*(s(1)+s(n-1))*...*(s(k)+s(k+1)))^2.
If n = 2*k, then s**s(n) has the form 2*s(k)*m^2, where m is an integer.
Table 5. Others
s = A000201, LW      t = A001950, UW         s**t = A374863
s = A000045, F       t = A000032, L          s**t = A374865
s = A005843, 2*n     t = A005408, 2*n+1      s**t = A085528, (2*n+1)^(n+1)
s = A016777, 3*n+1   t = A016789, 3*n+2      s**t = A091482
s = A005408, 2*n+1   t = A000045, F          s**t = A374867
s = A005408, 2*n+1   t = A000032, L          s**t = A374868
s = A005408, 2*n+1   t = A000079, 2^n        s**t = A374869
s = A000027, n       t = A000142, n!         s**t = A374871
s = A005408, 2*n+1   t = A000142, n!         s**t = A374872
s = A000079, 2^n     t = A000142, n!         s**t = A374874
s = A000142, n!      t = A000045, F          s**t = A374875
s = A000142, n!      t = A000032, L          s**t = A374876
s = A005408, 2*n+1   t = A016777, 3*n+1      s**t = A352601
s = A005408, 2*n+1   t = A016789, 3*n+2      s**t = A064352
Table 6. Arrays of coefficients of s(x)**t(x), where s(x) and t(x) are polynomials
s(x)              t(x)            s(x)**t(x)
 n                 x              A132393
n^2                x              A269944
x+1               x+1             A038220
x+2               x+2             A038244
 x                x+3             A038220
nx                x+1             A094638
 1              x^2+x+1           A336996
n^2 x             x+1             A375041
n^2 x            2x+1             A375042
n^2 x             x+2             A375043
2^n x             x+1             A375044
2^n              2x+1             A375045
2^n              x+2              A375046
x+1              F(n)             A375047
x+1             x+F(n)            A375048
x+F(n)          x+F(n)            A375049

			a(0) = 0 + 0 = 0
a(1) = (0+1) * (1+0) = 1
a(2) = (0+1) * (1+1) * (1+0) = 2
a(3) = (0+2) * (1+1) * (1+1) * (2+0) = 16
As noted above, a(2*k+1) is a square for k>=0. The first 5 squares are 1, 16, 3600, 12320100, 701841817600, with corresponding square roots 1, 4, 60, 3510, 837760.
If n = 2*k, then s**s(n) has the form 2*F(k)*m^2, where m is an integer and F(k) is the k-th Fibonacci number; e.g., a(6) = 2*F(3)*(192)^2.

Cf. A000045, A374850-to-A374881, A374884-to-A374894, A375041-to-A375049.

a:= n-> (F-> mul(F(n-j)+F(j), j=0..n))(combinat[fibonacci]):
seq(a(n), n=0..15);  # Alois P. Heinz, Aug 02 2024

s[n_] := Fibonacci[n]; t[n_] := Fibonacci[n];
u[n_] := Product[s[k] + t[n - k], {k, 0, n}];
Table[u[n], {n, 0, 20}]

a(n)=prod(k=0, n, fibonacci(k) + fibonacci(n-k)) \\ Andrew Howroyd, Jul 31 2024

a(n) ~ c * phi^(3*n^2/4 + n) / 5^((n+1)/2), where c = QPochhammer(-1, 1/phi^2)^2/2 if n is even and c = phi^(1/4) * QPochhammer(-phi, 1/phi^2)^2 / (phi + 1)^2 if n is odd, and phi = A001622 is the golden ratio. - Vaclav Kotesovec, Aug 01 2024

A085527 a(n) = (2n+1)^n.

Original entry on oeis.org

1, 3, 25, 343, 6561, 161051, 4826809, 170859375, 6975757441, 322687697779, 16679880978201, 952809757913927, 59604644775390625, 4052555153018976267, 297558232675799463481, 23465261991844685929951, 1977985201462558877934081, 177482997121587371826171875

Offset: 0

N. J. A. Sloane, Jul 05 2003

a(n) is the determinant of the zigzag matrix Z(n) (see A088961). - Paul Boddington, Nov 03 2003
a(n) is also the number of rho-labeled graphs with n edges. A graph with n edges is a rho-labeled graph if there exists a one-to-one mapping from its vertex set to {0,1,...,2n} such that every edge receives as a label the absolute difference of its end-vertices and the edge labels are x1,x2,...,xn where xi=i or xi=2n+1-i. - Christian Barrientos and Sarah Minion, Feb 20 2015
a(n) is the number of nodes in the canonical automaton for the affine Weyl group of types B_n and C_n. - Tom Edgar, May 12 2016
a(n) is the number of rooted (at an edge) 2-trees with n+2 edges. See also A052750. - Nikos Apostolakis, Dec 05 2018

Anders Björner and Francesco Brenti, Combinatorics of Coxeter groups. Graduate Texts in Mathematics, 231. Springer, New York, 2005.

G. C. Greubel, Table of n, a(n) for n = 0..350
Karola Mészáros, Labeling the Regions of the Type C_n Shi Arrangement, The Electronic Journal of Combinatorics, vol. 20, no. 2, (2013).
Zhi-Wei Sun, Fedor Petrov, A surprising identity, MathOverflow, Jan 17 2019.

Cf. A062971, A085528, A088961, A099753, A214406, A052750.

List([0..20],n->(2*n+1)^n); # Muniru A Asiru, Dec 05 2018

[(2*n+1)^n: n in [0..20]]; // Wesley Ivan Hurt, Mar 01 2015

A085527:=n->(2*n+1)^n: seq(A085527(n), n=0..20); # Wesley Ivan Hurt, Mar 01 2015

Table[(2 n + 1)^n, {n, 0, 20}] (* Wesley Ivan Hurt, Mar 01 2015 *)

PARI
```
a(n)=(2*n+1)^n;
```

def A085527(n): return ((n<<1)|1)**n # Chai Wah Wu, Nov 10 2024

E.g.f.: sqrt(2)/(2*(1+LambertW(-2*x))*sqrt(-x/LambertW(-2*x))). - Vladeta Jovovic, Oct 16 2004
For r = 0, 1, 2, ..., the e.g.f. for the sequence whose n-th term is (2*n+1)^(n+r) can be expressed in terms of the function U(z) = Sum_{n >= 0} (2*n+1)^(n-1)*z^(2*n+1)/(2^n*n!). See A214406 for details. In the present case, r = 0, and the resulting e.g.f. is 1/z*U(z)/(1 - U(z)^2) taken at z = sqrt(2*x). - Peter Bala, Aug 06 2012
a(n) = [x^n] 1/(1 - (2*n+1)*x). - Ilya Gutkovskiy, Oct 10 2017
a(n) = (-2)^n * D(2*n + 1), where D(n) is the determinant of the n X n matrix M with elements M(j, k) = cos(Pi*j*k/n). See the Zhi-Wei Sun, Petrov link. - Peter Luschny, Sep 19 2021
a(n) ~ exp(1/2) * 2^n * n^n. - Vaclav Kotesovec, Dec 05 2021
Series reversion of (1 - x)^2 * log(1/(1 - x)) begins  x + 3*x^2/2! + 25*x^3/3! + 343*x^4/4! + 6561*x^5/5! + .... - Peter Bala, Sep 27 2023
a(n) = Product_{k=1..n} tan(k*Pi/(1+2*n))^(2*n). - Chai Wah Wu, Nov 10 2024

A099753 a(n) = (2*n+1)^(n+2).

Original entry on oeis.org

1, 27, 625, 16807, 531441, 19487171, 815730721, 38443359375, 2015993900449, 116490258898219, 7355827511386641, 504036361936467383, 37252902984619140625, 2954312706550833698643, 250246473680347348787521, 22550116774162743178682911, 2154025884392726618070214209

Offset: 0

Kari Lajunen (Kari.Lajunen(AT)Welho.com), Nov 11 2004

G. C. Greubel, Table of n, a(n) for n = 0..345

Cf. A000169, A085527, A085528, A214406.

List([0..30], n-> (2*n+1)^(n+2)); # G. C. Greubel, Sep 03 2019

[(2*n+1)^(n+2): n in [0..30]]; // G. C. Greubel, Sep 03 2019

seq((2*n+1)^(n+2), n=0..30); # G. C. Greubel, Sep 03 2019

Table[(2*n+1)^(n+2), {n,0,30}] (* G. C. Greubel, Sep 03 2019 *)

vector(30, n, (2*n-1)^(n+1)) \\ G. C. Greubel, Sep 03 2019

[(2*n+1)^(n+2) for n in (0..30)] # G. C. Greubel, Sep 03 2019

From Peter Bala, Aug 06 2012: (Start)
E.g.f.: d^2/dx^2{(2*x/T(2*x))^(3/2)*1/(1 - T(2*x))} = 1 + 27*x + 625*x^2/2! + ..., where T(x) is the tree function sum {n >=1} n^(n-1)*x^n/n! of A000169.
For r = 0, 1, 2, ..., the e.g.f. for the sequence (2*n+1)^(n+r) can be expressed in terms of the function U(z) = sum {n >= 0} (2*n+1)^(n-1)*z^(2*n+1)/(2^n*n!). See A214406 for details. In the present case, r = 2, and the resulting e.g.f. is 1/z*U(z)*(1 + 8*U(z)^2 + 3*U(z)^4)/(1 - U(z)^2)^5 taken at z = sqrt(2*x).
(End)

Terms a(13) onward added by G. C. Greubel, Sep 03 2019

A085529 a(n) = (2n+1)^(2n+1).

Original entry on oeis.org

1, 27, 3125, 823543, 387420489, 285311670611, 302875106592253, 437893890380859375, 827240261886336764177, 1978419655660313589123979, 5842587018385982521381124421, 20880467999847912034355032910567, 88817841970012523233890533447265625, 443426488243037769948249630619149892803

Offset: 0

N. J. A. Sloane, Jul 05 2003

nonn

a(n) == 2*n + 1 (mod 24). - Mathew Englander, Aug 16 2020

Cf. A000312, A005408, A016754, A085527, A085528, A085530, A085531, A085532, A085533, A085534, A085535.

Cf. A073009, A083648.

#^#&/@Range[1,31,2] (* Harvey P. Dale, Oct 05 2019 *)

From Mathew Englander, Aug 16 2020: (Start)
a(n) = A000312(2*n + 1).
a(n) = A016754(n)^n * (2*n + 1).
a(n) = A085527(n)^2 * (2*n + 1).
a(n) = A085528(n)^2 / (2*n + 1).
a(n) = A085530(n) * A005408(n).
a(n) = A085531(n) * A016754(n).
a(n) = A085532(n)^2 - A215265(2*n + 1).
a(n) = A085533(n) + A045531(2*n + 1).
a(n) = A085534(n+1) - A007781(2*n + 1).
a(n) = A085535(n+1) - A055869(2*n + 1).
(End)
Sum_{n>=0} 1/a(n) = (A073009 + A083648)/2 = 1.0373582538... . - Amiram Eldar, May 17 2022

A165156 n^(2n-1)-(2n-1)^n.

Original entry on oeis.org

0, -1, 118, 13983, 1894076, 361025495, 96826261890, 35181809198207, 16677063111790072, 9999993868933742199, 7400249593980659558990, 6624737245034612579099807, 7056410013376700546645974068

Offset: 1

Vladimir Joseph Stephan Orlovsky, Sep 05 2009

sign

Cf. A085524, A085528

p=1;lst={};Do[AppendTo[lst,n^p-p^n];p=p+2,{n,4!}];lst
Table[n^(2n-1)-(2n-1)^n,{n,20}] (* Harvey P. Dale, Aug 10 2023 *)

a(n) = A085524(n)-A085528(n-1).

Definition corrected by R. J. Mathar, Sep 06 2009

A364870 Array read by ascending antidiagonals: A(n, k) = (n + k)^n, with k >= 0.

Original entry on oeis.org

1, 1, 1, 4, 2, 1, 27, 9, 3, 1, 256, 64, 16, 4, 1, 3125, 625, 125, 25, 5, 1, 46656, 7776, 1296, 216, 36, 6, 1, 823543, 117649, 16807, 2401, 343, 49, 7, 1, 16777216, 2097152, 262144, 32768, 4096, 512, 64, 8, 1, 387420489, 43046721, 4782969, 531441, 59049, 6561, 729, 81, 9, 1

Offset: 0

Stefano Spezia, Aug 11 2023

			The array begins:
     1,    1,     1,     1,     1,      1, ...
     1,    2,     3,     4,     5,      6, ...
     4,    9,    16,    25,    36,     49, ...
    27,   64,   125,   216,   343,    512, ...
   256,  625,  1296,  2401,  4096,   6561, ...
  3125, 7776, 16807, 32768, 59049, 100000, ...
  ...

Cf. A000012 (n=0), A000169, A000272, A000312 (k=0), A007830 (k=3), A008785 (k=4), A008786 (k=5), A008787 (k=6), A031973 (antidiagonal sums), A052746 (2nd superdiagonal), A052750, A062971 (main diagonal), A079901 (read by descending antidiagonals), A085527 (1st superdiagonal), A085528 (1st subdiagonal), A085532, A099753.

A[n_,k_]:=(n+k)^n; Join[{1},Table[A[n-k,k],{n,9},{k,0,n}]]//Flatten

E.g.f. of k-th column: LambertW(-x)^k/(x^k*(1 + LambertW(-x))).

cp's OEIS Frontend

A374848 Obverse convolution A000045**A000045; see Comments.

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A085527 a(n) = (2n+1)^n.

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A099753 a(n) = (2*n+1)^(n+2).

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A085529 a(n) = (2n+1)^(2n+1).

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A165156 n^(2*n-1)-(2*n-1)^n.

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A364870 Array read by ascending antidiagonals: A(n, k) = (n + k)^n, with k >= 0.

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A165156 n^(2n-1)-(2n-1)^n.