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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.

Showing 1-5 of 5 results.

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

A124320 Triangle read by rows: T(n,k) = k!*binomial(n+k-1,k) (n >= 0, 0 <= k <= n), rising factorial power, Pochhammer symbol.

Original entry on oeis.org

1, 1, 1, 1, 2, 6, 1, 3, 12, 60, 1, 4, 20, 120, 840, 1, 5, 30, 210, 1680, 15120, 1, 6, 42, 336, 3024, 30240, 332640, 1, 7, 56, 504, 5040, 55440, 665280, 8648640, 1, 8, 72, 720, 7920, 95040, 1235520, 17297280, 259459200, 1, 9, 90, 990, 11880, 154440, 2162160, 32432400, 518918400, 8821612800
Offset: 0

Views

Author

Emeric Deutsch, Oct 26 2006

Keywords

Comments

This is the Pochhammer function which is defined P(x,n) = x*(x+1)*...*(x+n-1). By convention P(0,0) = 1. Also known as the rising factorial power. - Peter Luschny, Jan 09 2011

Examples

			Triangle starts:
[0]  1;
[1]  1, 1;
[2]  1, 2,   6;
[3]  1, 3,  12,  60;
[4]  1, 4,  20, 120,  840;
[5]  1, 5,  30, 210, 1680, 15120;
[6]  1, 6,  42, 336, 3024, 30240, 332640;
[7]  1, 7,  56, 504, 5040, 55440, 665280, 8648640;
Array starts:
[0] 1,  1,   6,   60,   840,   15120,   332640,   8648640, ... A000407
[1] 1,  2,  12,  120,  1680,   30240,   665280,  17297280, ... A001813
[2] 1,  3,  20,  210,  3024,   55440,  1235520,  32432400, ... A006963
[3] 1,  4,  30,  336,  5040,   95040,  2162160,  57657600, ... A001761
[4] 1,  5,  42,  504,  7920,  154440,  3603600,  98017920, ... A102693
[5] 1,  6,  56,  720, 11880,  240240,  5765760, 160392960, ... A093197
[6] 1,  7,  72,  990, 17160,  360360,  8910720, 253955520, ... A203473
[7] 1,  8,  90, 1320, 24024,  524160, 13366080, 390700800, ...
[8] 1,  9, 110, 1716, 32760,  742560, 19535040, 586051200, ...
[9] 1, 10, 132, 2184, 43680, 1028160, 27907200, 859541760, ...

References

  • Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, Addison-Wesley, 1994.

Links

Crossrefs

Rows of array: A000407, A001813, A006963, A001761, A102693, A093197, A203473.
Cf. A123680 (row sums), A352601 (array main diagonal), A123680, A068424 (falling factorial power).

Programs

  • Maple
    T:=proc(n,k) if k<=n then binomial(n+k-1,k)*k! else 0 fi end: for n from 0 to 9 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form
A124320 := (n,k)-> `if`(n=0 and k=0,1,pochhammer(n,k)); seq(print(seq(A124320(n,k),k=0..n)),n=0..5); # Peter Luschny, Jan 09 2011
  • Mathematica
    Table[Pochhammer[n,k], {n,0,5},{k,0,n}]//Flatten (* Peter Luschny, Jan 09 2011 *)
  • PARI
    for(n=0,10, for(k=0,n, print1(if(n==0 && k==0, 1, (n+k-1)!/(n-1)!), ", "))) \\ G. C. Greubel, Nov 19 2017
  • Sage
    for n in (0..5) : [rising_factorial(n, k) for k in (0..n)] # Peter Luschny, Jan 09 2011

Formula

T(n,k) = GAMMA(n+k)/GAMMA(n) for n>0. - Peter Luschny, Jan 09 2011

A384164 a(n) = Product_{k=0..n-1} (3*n+k).

Original entry on oeis.org

1, 3, 42, 990, 32760, 1395360, 72681840, 4475671200, 318073392000, 25622035084800, 2306992893004800, 229601607198163200, 25028504609870361600, 2965681982933429760000, 379534960108578193920000, 52170410224819317150720000, 7666009844358186506465280000, 1199151678674216896627654656000
Offset: 0

Views

Author

Seiichi Manyama, May 21 2025

Keywords

Links

Crossrefs

Cf. A384164, A384165.
Cf. A000407, A352601.
Cf. A061924, A124320.

Programs

  • Magma
    [1] cat  [&*[(3*n + k): k in [0..n-1]]: n in [1..16]]; // Vincenzo Librandi, May 22 2025
  • Mathematica
    a[n_]:=Product[(3*n+k),{k,0,n-1}]; Table[a[n],{n,0,15}] (* Vincenzo Librandi, May 22 2025 *)
  • PARI
    a(n) = prod(k=0, n-1, 3*n+k);
  • Python
    from sympy import rf
def A384164(n): return rf(3*n,n) # Chai Wah Wu, May 21 2025
  • Sage
    def a(n): return rising_factorial(3*n, n)

Formula

a(n) = RisingFactorial(3*n,n) = A124320(3*n,n) = n! * binomial(4*n-1,n).
a(n) = n! * [x^n] 1/(1 - x)^(3*n).
a(n) = (3/4) * A061924(n) for n > 0.

A384163 a(n) = Product_{k=0..n-1} (2*n+3*k).

Original entry on oeis.org

1, 2, 28, 648, 20944, 869440, 44089920, 2641533440, 182573036800, 14299419214080, 1251598943795200, 121073405444992000, 12826824167930572800, 1477015178613438464000, 183679785389526871244800, 24533610049517447983104000, 3502810763000490499317760000, 532374290389646285405913088000
Offset: 0

Views

Author

Seiichi Manyama, May 21 2025

Keywords

Links

Crossrefs

Cf. A352601.

Programs

  • Magma
    [1] cat  [&*[(2*n+3*k): k in [0..n-1]]: n in [1..16]]; // Vincenzo Librandi, May 22 2025
  • Mathematica
    a[n_]:=Product[(2*n+3*k),{k,0,n-1}]; Table[a[n],{n,0,15}] (* Vincenzo Librandi, May 22 2025 *)
  • PARI
    a(n) = prod(k=0, n-1, 2*n+3*k);
  • Sage
    def a(n): return 3^n*rising_factorial(2*n/3, n)

Formula

a(n) = 3^n * RisingFactorial(2*n/3,n).
a(n) = n! * [x^n] 1/(1 - 3*x)^(2*n/3).
a(n) = (2/5) * 3^n * n! * binomial(5*n/3,n) for n > 0.

A384261 a(n) = Product_{k=0..n-1} (2*n+k-1).

Original entry on oeis.org

1, 1, 12, 210, 5040, 154440, 5765760, 253955520, 12893126400, 741354768000, 47621141568000, 3379847863392000, 262662462526464000, 22183557976419840000, 2023140487449489408000, 198155371076302768128000, 20744817468539834621952000, 2311708772421640603275264000
Offset: 0

Views

Author

Seiichi Manyama, May 23 2025

Keywords

Crossrefs

Cf. A064352, A352601.
Cf. A090816, A117671.

Programs

  • PARI
    a(n) = prod(k=0, n-1, 2*n+k-1);
  • Python
    from sympy import rf
def a(n): return rf(2*n-1, n)
  • Sage
    def a(n): return rising_factorial(2*n-1, n)

Formula

a(n) = RisingFactorial(2*n-1,n).
a(n) = n! * [x^n] 1/(1 - x)^(2*n-1).
a(n) = n! * binomial(3*n-2,n).
D-finite with recurrence 2*(-2*n+3)*a(n) +3*(3*n-2)*(3*n-4)*a(n-1)=0. - R. J. Mathar, May 26 2025
Showing 1-5 of 5 results.

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