cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A374848 Obverse convolution A000045**A000045; see Comments.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jul 31 2024

Keywords

Comments

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

Examples

			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.

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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}]
  • PARI
    a(n)=prod(k=0, n, fibonacci(k) + fibonacci(n-k)) \\ Andrew Howroyd, Jul 31 2024

Formula

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

A369674 a(n) = Product_{k=0..n} (3^k + 3^(n-k)).

Original entry on oeis.org

2, 16, 600, 112896, 108928800, 544431476736, 14105702277360000, 1900051576637594075136, 1328360485647389567734080000, 4830166933124609654538067824869376, 91168969237139220357818392868757600000000, 8950497893393998236587417126220897399198550327296
Offset: 0

Views

Author

Paul D. Hanna, Feb 06 2024

Keywords

Examples

			a(0) = (1 + 1) = 2;
a(1) = (1 + 3)*(3 + 1) = 16;
a(2) = (1 + 3^2)*(3 + 3)*(3^2 + 1) = 600;
a(3) = (1 + 3^3)*(3 + 3^2)*(3^2 + 3)*(3^3 + 1) = 112896;
a(4) = (1 + 3^4)*(3 + 3^3)*(3^2 + 3^2)*(3^3 + 3)*(3^4 + 1) = 108928800;
a(5) = (1 + 3^5)*(3 + 3^4)*(3^2 + 3^3)*(3^3 + 3^2)*(3^4 + 3)*(3^5 + 1) = 544431476736;
...
RELATED SERIES.
Let F(x) be the g.f. of A369557, then
F(1/3) = 2 + 16/3^2 + 600/3^6 + 112896/3^12 + 108928800/3^20 + 544431476736/3^30 + 14105702277360000/3^42 + ... + a(n)/3^(n*(n+1)) + ... = 4.847274134844057155467506697748724715389597193...

Crossrefs

Cf. A369673, A369675, A369676, A369557.

Programs

  • PARI
    {a(n) = prod(k=0, n, 3^k + 3^(n-k))}
for(n=0, 15, print1(a(n), ", "))

Formula

a(n) = Product_{k=0..n} (3^k + 3^(n-k)).
a(n) = 3^(n*(n+1)) * Product_{k=0..n} (1/3^k + 1/3^(n-k)).
a(n) = 3^(n*(n+1)/2) * Product_{k=0..n} (1 + 1/3^(n-2*k)).
From Vaclav Kotesovec, Feb 07 2024: (Start)
a(n) ~ c * 3^(3*n^2/4 + n), where
c = 2.538295806020848... = QPochhammer(-1, 1/9)^2/2 if n is even and
c = 2.539569717896307... = 3^(1/4) * QPochhammer(-3, 1/9)^2 / 16 if n is odd. (End)

A369675 a(n) = Product_{k=0..n} (4^k + 4^(n-k)).

Original entry on oeis.org

2, 25, 2312, 1690000, 9773138432, 454542400000000, 167983232813812416512, 499835663627223040000000000, 11821129880009981801801971612516352, 2251076882713432721110048178176000000000000, 3407215210591493267547957182357614317126952945713152, 41525058946342607360045945411073338768005424742400000000000000
Offset: 0

Views

Author

Paul D. Hanna, Feb 06 2024

Keywords

Examples

			a(0) = (1 + 1) = 2;
a(1) = (1 + 4)*(4 + 1) = 25;
a(2) = (1 + 4^2)*(4 + 4)*(4^2 + 1) = 2312;
a(3) = (1 + 4^3)*(4 + 4^2)*(4^2 + 4)*(4^3 + 1) = 1690000;
a(4) = (1 + 4^4)*(4 + 4^3)*(4^2 + 4^2)*(4^3 + 4)*(4^4 + 1) = 9773138432;
a(5) = (1 + 4^5)*(4 + 4^4)*(4^2 + 4^3)*(4^3 + 4^2)*(4^4 + 4)*(4^5 + 1) = 454542400000000;
...
RELATED SERIES.
Let F(x) be the g.f. of A369557, then
F(1/4) = 2 + 25/4^2 + 2312/4^6 + 1690000/4^12 + 9773138432/4^20 + 454542400000000/4^30 + ... + a(n)/4^(n*(n+1)) + ... = 4.236976626306045459467696438142250301516563681...

Crossrefs

Cf. A369673, A369674, A369676, A369557.

Programs

  • PARI
    {a(n) = prod(k=0, n, 4^k + 4^(n-k))}
for(n=0, 15, print1(a(n), ", "))

Formula

a(n) = Product_{k=0..n} (4^k + 4^(n-k)).
a(n) = 4^(n*(n+1)) * Product_{k=0..n} (1/4^k + 1/4^(n-k)).
a(n) = 4^(n*(n+1)/2) * Product_{k=0..n} (1 + 1/4^(n-2*k)).
From Vaclav Kotesovec, Feb 07 2024: (Start)
a(n) ~ c * 4^(3*n^2/4 + n), where
c = 2.276671433133289... = QPochhammer(-1, 1/16)^2/2 if n is even and
c = 2.284052876870834... = sqrt(2) * QPochhammer(-4, 1/16)^2 / 25 if n is odd. (End)

A369676 a(n) = Product_{k=0..n} (5^k + 5^(n-k)).

Original entry on oeis.org

2, 36, 6760, 14288400, 331135220000, 87265295649000000, 252668462115852250000000, 8322480168806663555062500000000, 3012058207750727786980181328125000000000, 12401474551899042876552569922821191406250000000000, 561039675887726306551826113078284190093383789062500000000000
Offset: 0

Views

Author

Paul D. Hanna, Feb 06 2024

Keywords

Comments

From Vaclav Kotesovec, Feb 07 2024: (Start)
For q > 1, Product_{k=0..n} (q^k + q^(n-k)) ~ c * q^(3*n^2/4 + n), where
c = QPochhammer(-1, 1/q^2)^2/2 if n is even and
c = q^(1/4) * QPochhammer(-q, 1/q^2)^2 / (q + 1)^2 if n is odd. (End)

Examples

			a(0) = (1 + 1) = 2;
a(1) = (1 + 5)*(5 + 1) = 36;
a(2) = (1 + 5^2)*(5 + 5)*(5^2 + 1) = 6760;
a(3) = (1 + 5^3)*(5 + 5^2)*(5^2 + 5)*(5^3 + 1) = 14288400;
a(4) = (1 + 5^4)*(5 + 5^3)*(5^2 + 5^2)*(5^3 + 5)*(5^4 + 1) = 331135220000;
a(5) = (1 + 5^5)*(5 + 5^4)*(5^2 + 5^3)*(5^3 + 5^2)*(5^4 + 5)*(5^5 + 1) = 87265295649000000;
...
RELATED SERIES.
Let F(x) be the g.f. of A369557, then
F(1/5) = 2 + 36/5^2 + 6760/5^6 + 14288400/5^12 + 331135220000/5^20 + 87265295649000000/5^30 + ... + a(n)/5^(n*(n+1)) + ... = 3.934732308501055907377639201049737298238369356...

Crossrefs

Cf. A369673, A369674, A369675, A369557.

Programs

  • PARI
    {a(n) = prod(k=0, n, 5^k + 5^(n-k))}
for(n=0, 15, print1(a(n), ", "))

Formula

a(n) = Product_{k=0..n} (5^k + 5^(n-k)).
a(n) = 5^(n*(n+1)) * Product_{k=0..n} (1/5^k + 1/5^(n-k)).
a(n) = 5^(n*(n+1)/2) * Product_{k=0..n} (1 + 1/5^(n-2*k)).
From Vaclav Kotesovec, Feb 07 2024: (Start)
a(n) ~ c * 5^(3*n^2/4 + n), where
c = 2.170417138549358... = QPochhammer(-1, 1/25)^2/2 if n is even and
c = 2.189351749288445... = 5^(1/4) * QPochhammer(-5, 1/25)^2 / 36 if n is odd. (End)

A369557 Expansion of Sum_{n>=0} Product_{k=0..n} (x^k + x^(n-k)).

Original entry on oeis.org

3, 4, 2, 6, 3, 4, 9, 4, 8, 6, 13, 8, 12, 12, 10, 22, 13, 22, 14, 26, 20, 34, 23, 32, 36, 34, 42, 36, 59, 38, 67, 46, 75, 56, 82, 66, 98, 84, 100, 102, 105, 126, 116, 152, 119, 184, 136, 202, 154, 230, 181, 256, 203, 276, 250, 306, 285, 326, 342, 348, 398, 374, 463, 404, 525, 438, 610, 486, 666, 542, 744, 610
Offset: 0

Views

Author

Paul D. Hanna, Feb 06 2024

Keywords

Examples

			G.f.: A(x) = 3 + 4*x + 2*x^2 + 6*x^3 + 3*x^4 + 4*x^5 + 9*x^6 + 4*x^7 + 8*x^8 + 6*x^9 + 13*x^10 + 8*x^11 + 12*x^12 + ...
where
A(x) = (1 + 1) + (1 + x)*(x + 1) + (1 + x^2)*(x + x)*(x^2 + 1) + (1 + x^3)*(x + x^2)*(x^2 + x)*(x^3 + 1) + (1 + x^4)*(x + x^3)*(x^2 + x^2)*(x^3 + x)*(x^4 + 1) + (1 + x^5)*(x + x^4)*(x^2 + x^3)*(x^3 + x^2)*(x^4 + x)*(x^5 + 1) + ...
Also,
A(1/x) = (1 + 1) + (1 + x)*(x + 1)/x^2 + (1 + x^2)*(x + x)*(x^2 + 1)/x^6 + (1 + x^3)*(x + x^2)*(x^2 + x)*(x^3 + 1)/x^12 + (1 + x^4)*(x + x^3)*(x^2 + x^2)*(x^3 + x)*(x^4 + 1)/x^20 + (1 + x^5)*(x + x^4)*(x^2 + x^3)*(x^3 + x^2)*(x^4 + x)*(x^5 + 1)/x^30 + ...
For example, at x = 1/2,
A(1/2) = 2 + 9/2^2 + 100/2^6 + 2916/2^12 + 231200/2^20 + 50808384/2^30 + 31258240000/2^42 + 54112148361216/2^56 + 264265663201280000/2^72 + ... + A369673(n)/2^(n*(n+1)) + ... = 6.80013983505192354264...
SPECIFIC VALUES.
A(t) = 4 at t = 0.21135479438007733067820905390237206358880...
A(t) = 5 at t = 0.35111207737762337157349938790010474080253...
A(t) = 6 at t = 0.44509902476179757380223857309576063477813...
A(3/4) = 18.04139246037655138841324835985762487898724341...
A(2/3) = 11.59103511448176661974748662249737201844158309...
A(Phi) = 9.595623356758087506923478384122062088751068609...
A(1/2) = 6.800139835051923542641455169580774467247971025...
A(1/3) = 4.847274134844057155467506697748724715389597193...
A(1/4) = 4.236976626306045459467696438142250301516563681...
A(1/5) = 3.934732308501055907377639201049737298238369356...

Links

Crossrefs

Cf. A369673, A369674, A369675, A369676, A376542, A376621.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Sum[Product[x^j + x^(k - j), {j, 0, k}], {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 08 2024 *)
nmax = 100; CoefficientList[Series[-1 + 2*Sum[x^(k^2) * Product[1 + x^(2*j), {j, 1, k}]^2, {k, 0, Sqrt[nmax]}] + Sum[x^((k-1)*k) * Product[1 + x^(2*j-1), {j, 1, k}]^2, {k, 0, Sqrt[nmax] + 1}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 27 2024 *)
  • PARI
    {a(n) = my(A = sum(m=0,n+1, prod(k=0,m, x^k + x^(m-k)) +x*O(x^n) )); polcoeff(A,n)}
for(n=0,70, print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = Sum_{n>=0} Product_{k=0..n} (x^k + x^(n-k)).
(2) A(x) = Sum_{n>=0} x^(n*(n+1)) * Product_{k=0..n} (1/x^k + 1/x^(n-k)).
(3) A(x) = Sum_{n>=0} x^(n*(n+1)/2) * Product_{k=0..n} (1 + x^(n-2*k)).
From Vaclav Kotesovec, Sep 29 2024: (Start)
a(n) ~ c * d^sqrt(n) / sqrt(n), where d = A376621 = 2.7510850908889199... and c = sqrt((1 + ((197 - sqrt(27/31)) / 62)^(1/3) + ((197 + sqrt(27/31)) / 62)^(1/3))/3) = 1.146046709280363...
a(n) ~ 4*A376542(n). (End)

A369680 a(n) = Product_{k=0..n} (2^k + 3^(n-k)).

Original entry on oeis.org

2, 12, 250, 19404, 5780918, 6691848108, 30261978906250, 535757771934053916, 37171553237849766044342, 10113067879819381109893992732, 10789224041146220828897229003906250, 45150513047221188662211059385153001179564, 741117672560101894851755994230829254062662140918
Offset: 0

Views

Author

Paul D. Hanna, Feb 06 2024

Keywords

Comments

For p > 1, q > 1, limit_{n->oo} (Product_{k=0..n} (p^k + q^(n-k)) )^(1/n^2) = q^(1/2)*p^(1/(2*(1 + log(q)/log(p)))) = p^(1/2)*q^(1/(2*(1 + log(p)/log(q)))). - Vaclav Kotesovec, Feb 07 2024
Equivalently, limit_{n->oo} ( Product_{k=0..n} (p^k + q^(n-k)) )^(1/n^2) = exp((1/2) * (log(p)^2 + log(p)*log(q) + log(q)^2) / log(p*q)). - Paul D. Hanna, Feb 08 2024

Examples

			a(0) = (1 + 1) = 2;
a(1) = (1 + 3)*(2 + 1) = 12;
a(2) = (1 + 3^2)*(2 + 3)*(2^2 + 1) = 250;
a(3) = (1 + 3^3)*(2 + 3^2)*(2^2 + 3)*(2^3 + 1) = 19404;
a(4) = (1 + 3^4)*(2 + 3^3)*(2^2 + 3^2)*(2^3 + 3)*(2^4 + 1) = 5780918;
a(5) = (1 + 3^5)*(2 + 3^4)*(2^2 + 3^3)*(2^3 + 3^2)*(2^4 + 3)*(2^5 + 1) = 6691848108;
...
RELATED SERIES.
Sum_{n>=0} Product_{k=0..n} (1/2^k + 1/3^(n-k)) = 2 + 12/6 + 250/6^3 + 19404/6^6 + 5780918/6^10 + 6691848108/6^15 + ... + a(n)/6^(n*(n+1)/2) + ... = 5.6846111010137973166832330595516662115250385271...

Crossrefs

Cf. A369673, A369674, A369675, A369676, A369677, A369678, A369679.
Cf. A323715.

Programs

  • Mathematica
    Table[Product[2^k+3^(n-k),{k,0,n}],{n,0,12}] (* James C. McMahon, Feb 07 2024 *)
  • PARI
    {a(n) = prod(k=0, n, 2^k + 3^(n-k))}
for(n=0, 15, print1(a(n), ", "))

Formula

a(n) = Product_{k=0..n} (2^k + 3^(n-k)).
a(n) = 6^(n*(n+1)/2) * Product_{k=0..n} (1/2^k + 1/3^(n-k)).
a(n) = 3^(n*(n+1)/2) * Product_{k=0..n} (1 + 2^n/6^k).
a(n) = 2^(n*(n+1)/2) * Product_{k=0..n} (1 + 3^n/6^k).
a(n) = 2^(-n*(n+1)/2) * Product_{k=0..n} (2^n + 6^k).
a(n) = 3^(-n*(n+1)/2) * Product_{k=0..n} (3^n + 6^k).
a(n) = 2^(n*(n+1)/2)*QPochhammer(-3^n, 1/6, n + 1). - Stefano Spezia, Feb 07 2024
Limit_{n->oo} a(n)^(1/n^2) = 2^(1/(2*(1 + log(3)/log(2)))) * sqrt(3) = 3^(1/(2*(1 + log(2)/log(3)))) * sqrt(2) = 1.9805589654474717155611061670180902111915926... - Vaclav Kotesovec, Feb 07 2024
Equivalently, limit_{n->oo} a(n)^(1/n^2) = exp((1/2) * (log(2)^2 + log(2)*log(3) + log(3)^2) / log(6)). - Paul D. Hanna, Feb 08 2024

A369677 a(n) = Product_{k=0..n} (2^k + 5^(n-k)).

Original entry on oeis.org

2, 18, 910, 275562, 509528318, 5782203860202, 403066704971309470, 172986911139059942455818, 457494980583771669025834718462, 7445459859979605380607238308201858858, 746155118699551878624986638597659812003763550, 461066589238234272286243169377378506495126815749310922
Offset: 0

Views

Author

Paul D. Hanna, Feb 07 2024

Keywords

Examples

			a(0) = (1 + 1) = 2;
a(1) = (1 + 5)*(2 + 1) = 18;
a(2) = (1 + 5^2)*(2 + 5)*(2^2 + 1) = 910;
a(3) = (1 + 5^3)*(2 + 5^2)*(2^2 + 5)*(2^3 + 1) = 275562;
a(4) = (1 + 5^4)*(2 + 5^3)*(2^2 + 5^2)*(2^3 + 5)*(2^4 + 1) = 509528318;
a(5) = (1 + 5^5)*(2 + 5^4)*(2^2 + 5^3)*(2^3 + 5^2)*(2^4 + 5)*(2^5 + 1) = 5782203860202;
...
RELATED SERIES.
Sum_{n>=0} Product_{k=0..n} (1/2^k + 1/5^(n-k)) = 2 + 18/10 + 910/10^3 + 275562/10^6 + 509528318/10^10 + 5782203860202/10^15 + ... + a(n)/10^(n*(n+1)/2) + ... = 5.0427178660718059961260933841217518099...

Crossrefs

Cf. A369673, A369674, A369675, A369676, A369678, A369679, A369680.

Programs

  • PARI
    {a(n) = prod(k=0, n, 2^k + 5^(n-k))}
for(n=0, 15, print1(a(n), ", "))

Formula

a(n) = Product_{k=0..n} (2^k + 5^(n-k)).
a(n) = 10^(n*(n+1)/2) * Product_{k=0..n} (1/2^k + 1/5^(n-k)).
a(n) = 5^(n*(n+1)/2) * Product_{k=0..n} (1 + 2^n/10^k).
a(n) = 2^(n*(n+1)/2) * Product_{k=0..n} (1 + 5^n/10^k).
a(n) = 2^(-n*(n+1)/2) * Product_{k=0..n} (2^n + 10^k).
a(n) = 5^(-n*(n+1)/2) * Product_{k=0..n} (5^n + 10^k).
a(n) = 2^(n*(n+1)/2)*QPochhammer(-5^n, 1/10, n + 1). - Stefano Spezia, Feb 07 2024
Limit_{n->oo} a(n)^(1/n^2) = 2^(1/(2*(1 + log(5)/log(2)))) * sqrt(5) = 5^(1/(2*(1 + log(2)/log(5)))) * sqrt(2) = 2.481958590195459039209137154563963236753327... - Vaclav Kotesovec, Feb 07 2024
Equivalently, limit_{n->oo} a(n)^(1/n^2) = exp((1/2) * (log(2)^2 + log(2)*log(5) + log(5)^2) / log(10)). - Paul D. Hanna, Feb 08 2024

A369678 a(n) = Product_{k=0..n} (3^k + 5^(n-k)).

Original entry on oeis.org

2, 24, 2080, 1382976, 7148699648, 287041728769536, 90391546425391144960, 221202979125273147766738944, 4237647337376998325597017538035712, 633933934421036224259931934460116571357184, 738292285249623417870561091674252758908330993254400
Offset: 0

Views

Author

Paul D. Hanna, Feb 07 2024

Keywords

Examples

			a(0) = (1 + 1) = 2;
a(1) = (1 + 5)*(3 + 1) = 24;
a(2) = (1 + 5^2)*(3 + 5)*(3^2 + 1) = 2080;
a(3) = (1 + 5^3)*(3 + 5^2)*(3^2 + 5)*(3^3 + 1) = 1382976;
a(4) = (1 + 5^4)*(3 + 5^3)*(3^2 + 5^2)*(3^3 + 5)*(3^4 + 1) = 7148699648;
a(5) = (1 + 5^5)*(3 + 5^4)*(3^2 + 5^3)*(3^3 + 5^2)*(3^4 + 5)*(3^5 + 1) = 287041728769536;
...
RELATED SERIES.
Sum_{n>=0} Product_{k=0..n} (1/3^k + 1/5^(n-k)) = 2 + 24/15 + 2080/15^3 + 1382976/15^6 + 7148699648/15^10 + 287041728769536/15^15 + ... + a(n)/15^(n*(n+1)/2) + ... = 4.3507806549816093424129450104392682482776...

Crossrefs

Cf. A369673, A369674, A369675, A369676, A369677, A369679, A369680.

Programs

  • PARI
    {a(n) = prod(k=0, n, 3^k + 5^(n-k))}
for(n=0, 15, print1(a(n), ", "))

Formula

a(n) = Product_{k=0..n} (3^k + 5^(n-k)).
a(n) = 15^(n*(n+1)/2) * Product_{k=0..n} (1/3^k + 1/5^(n-k)).
a(n) = 5^(n*(n+1)/2) * Product_{k=0..n} (1 + 3^n/15^k).
a(n) = 3^(n*(n+1)/2) * Product_{k=0..n} (1 + 5^n/15^k).
a(n) = 3^(-n*(n+1)/2) * Product_{k=0..n} (3^n + 15^k).
a(n) = 5^(-n*(n+1)/2) * Product_{k=0..n} (5^n + 15^k).
a(n) = 3^(n*(n+1)/2)*QPochhammer(-5^n, 1/15, n + 1). - Stefano Spezia, Feb 07 2024
Limit_{n->oo} a(n)^(1/n^2) = 3^(1/(2*(1 + log(5)/log(3)))) * sqrt(5) = 5^(1/(2*(1 + log(3)/log(5)))) * sqrt(3) = 2.794249622709633938040980858655052416325961... - Vaclav Kotesovec, Feb 07 2024
Equivalently, limit_{n->oo} a(n)^(1/n^2) = exp((1/2) * (log(3)^2 + log(3)*log(5) + log(5)^2) / log(15)). - Paul D. Hanna, Feb 08 2024

A369679 a(n) = Product_{k=0..n} (3^k + 4^(n-k)).

Original entry on oeis.org

2, 20, 1190, 449540, 1094267450, 17283181758500, 1774660248586902950, 1182579046508766038251700, 5134581376819479940742838299450, 144547890423248529154421336209389168500, 26527720524980501045637796065988864058001683750, 31574745853363739268697794406696294745967395732336262500
Offset: 0

Views

Author

Paul D. Hanna, Feb 07 2024

Keywords

Examples

			a(0) = (1 + 1) = 2;
a(1) = (1 + 4)*(3 + 1) = 20;
a(2) = (1 + 4^2)*(3 + 4)*(3^2 + 1) = 1190;
a(3) = (1 + 4^3)*(3 + 4^2)*(3^2 + 4)*(3^3 + 1) = 449540;
a(4) = (1 + 4^4)*(3 + 4^3)*(3^2 + 4^2)*(3^3 + 4)*(3^4 + 1) = 1094267450;
a(5) = (1 + 4^5)*(3 + 4^4)*(3^2 + 4^3)*(3^3 + 4^2)*(3^4 + 4)*(3^5 + 1) = 17283181758500;
...
RELATED SERIES.
Sum_{n>=0} Product_{k=0..n} (1/3^k + 1/4^(n-k)) = 2 + 20/12 + 1190/12^3 + 449540/12^6 + 1094267450/12^10 + 17283181758500/12^15 + ... + a(n)/12^(n*(n+1)/2) + ... = 4.5247082137580440222914164418070212438323...

Crossrefs

Cf. A369673, A369674, A369675, A369676, A369677, A369678, A369680.

Programs

  • PARI
    {a(n) = prod(k=0, n, 3^k + 4^(n-k))}
for(n=0, 15, print1(a(n), ", "))

Formula

a(n) = Product_{k=0..n} (3^k + 4^(n-k)).
a(n) = 12^(n*(n+1)/2) * Product_{k=0..n} (1/3^k + 1/4^(n-k)).
a(n) = 4^(n*(n+1)/2) * Product_{k=0..n} (1 + 3^n/12^k).
a(n) = 3^(n*(n+1)/2) * Product_{k=0..n} (1 + 4^n/12^k).
a(n) = 3^(-n*(n+1)/2) * Product_{k=0..n} (3^n + 12^k).
a(n) = 4^(-n*(n+1)/2) * Product_{k=0..n} (4^n + 12^k).
a(n) = 3^(n*(n+1)/2)*QPochhammer(-4^n, 1/12, n + 1). - Stefano Spezia, Feb 07 2024
Limit_{n->oo} a(n)^(1/n^2) = 3^(1/(2*(1 + log(4)/log(3)))) * 2 = 2^(1/(1 + log(3)/log(4))) * sqrt(3) = 2.54977004574388327607102436919328599299374003... - Vaclav Kotesovec, Feb 07 2024
Equivalently, limit_{n->oo} a(n)^(1/n^2) = exp((1/2) * (log(3)^2 + log(3)*log(4) + log(4)^2) / log(12)). - Paul D. Hanna, Feb 08 2024

A369681 a(n) = Product_{k=0..n} (4^k + 5^(n-k)).

Original entry on oeis.org

2, 30, 3978, 4987710, 58712437962, 6601051349841150, 7017151861981535193738, 70966047508527496843460412990, 6820716704126571481897874317127918922, 6205644698427009393117687864650447521113942270, 53916867047490616763228279441645027173409633988839675658
Offset: 0

Views

Author

Paul D. Hanna, Feb 07 2024

Keywords

Comments

For p > 1, q > 1, limit_{n->oo} ( Product_{k=0..n} (p^k + q^(n-k)) )^(1/n^2) = exp((1/2) * (log(p)^2 + log(p)*log(q) + log(q)^2) / log(p*q)); formula due to Vaclav Kotesovec (cf. A369680).

Examples

			a(0) = (1 + 1) = 2;
a(1) = (1 + 5)*(4 + 1) = 30;
a(2) = (1 + 5^2)*(4 + 5)*(4^2 + 1) = 3978;
a(3) = (1 + 5^3)*(4 + 5^2)*(4^2 + 5)*(4^3 + 1) = 4987710;
a(4) = (1 + 5^4)*(4 + 5^3)*(4^2 + 5^2)*(4^3 + 5)*(4^4 + 1) = 58712437962;
a(5) = (1 + 5^5)*(4 + 5^4)*(4^2 + 5^3)*(4^3 + 5^2)*(4^4 + 5)*(4^5 + 1) = 6601051349841150;
...
RELATED SERIES.
Sum_{n>=0} Product_{k=0..n} (1/4^k + 1/5^(n-k)) = 2 + 30/20 + 3978/20^3 + 4987710/20^6 + 58712437962/20^10 + 6601051349841150/20^15 + ... + a(n)/20^(n*(n+1)/2) + ... = 4.0811214259450988699292249336017494522520...

Crossrefs

Cf. A369673, A369674, A369675, A369676, A369677, A369678, A369679, A369680.

Programs

  • PARI
    {a(n) = prod(k=0, n, 4^k + 5^(n-k))}
for(n=0, 15, print1(a(n), ", "))

Formula

a(n) = Product_{k=0..n} (4^k + 5^(n-k)).
a(n) = 20^(n*(n+1)/2) * Product_{k=0..n} (1/4^k + 1/5^(n-k)).
a(n) = 5^(n*(n+1)/2) * Product_{k=0..n} (1 + 4^n/20^k).
a(n) = 4^(n*(n+1)/2) * Product_{k=0..n} (1 + 5^n/20^k).
a(n) = 4^(-n*(n+1)/2) * Product_{k=0..n} (4^n + 20^k).
a(n) = 5^(-n*(n+1)/2) * Product_{k=0..n} (5^n + 20^k).
Limit_{n->oo} a(n)^(1/n^2) = exp((1/2) * (log(4)^2 + log(4)*log(5) + log(5)^2) / log(20)) = 3.0816872899745614612763875038173884057052077... [from a formula by Vaclav Kotesovec].
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