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.

Showing 1-5 of 5 results.

A370419 A(n, k) = 2^n*Pochhammer(k/2, n). Square array read by ascending antidiagonals.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 3, 2, 1, 0, 15, 8, 3, 1, 0, 105, 48, 15, 4, 1, 0, 945, 384, 105, 24, 5, 1, 0, 10395, 3840, 945, 192, 35, 6, 1, 0, 135135, 46080, 10395, 1920, 315, 48, 7, 1, 0, 2027025, 645120, 135135, 23040, 3465, 480, 63, 8, 1
Offset: 0

Views

Author

Peter Luschny, Mar 04 2024

Keywords

Examples

			The array starts:
[0] 1,   1,    1,     1,     1,     1,     1,      1,      1, ...
[1] 0,   1,    2,     3,     4,     5,     6,      7,      8, ...
[2] 0,   3,    8,    15,    24,    35,    48,     63,     80, ...
[3] 0,  15,   48,   105,   192,   315,   480,    693,    960, ...
[4] 0, 105,  384,   945,  1920,  3465,  5760,   9009,  13440, ...
[5] 0, 945, 3840, 10395, 23040, 45045, 80640, 135135, 215040, ...
.
Seen as the triangle T(n, k) = A(n - k, k):
[0] 1;
[1] 0,   1;
[2] 0,   1,   1;
[3] 0,   3,   2,   1;
[4] 0,  15,   8,   3,  1;
[5] 0, 105,  48,  15,  4, 1;
[6] 0, 945, 384, 105, 24, 5, 1;
.
From _Werner Schulte_, Mar 07 2024: (Start)
Illustrating the LU decomposition of A:
    / 1                \   / 1 1 1 1 1 ... \   / 1   1   1   1    1 ... \
    | 0   1            |   |   1 2 3 4 ... |   | 0   1   2   3    4 ... |
    | 0   3   2        | * |     1 3 6 ... | = | 0   3   8  15   24 ... |
    | 0  15  18   6    |   |       1 4 ... |   | 0  15  48 105  192 ... |
    | 0 105 174 108 24 |   |         1 ... |   | 0 105 384 945 1920 ... |
    | . . .            |   | . . .         |   | . . .                  |. (End)

Links

Crossrefs

Columns: A000007, A001147, A000165, A001147 (shifted), A002866, A051577, A051578, A051579, A051580.
Rows: A000012, A001477, A005563, A370912, A190577.
Cf. A035342, A265649, A370890, A370982 (row sums of the triangle), A370915, A371025, A371077.

Programs

  • Maple
    A := (n, k) -> 2^n*pochhammer(k/2, n):
for n from 0 to 5 do seq(A(n, k), k = 0..9) od;
T := (n, k) -> A(n - k, k): seq(seq(T(n, k), k = 0..n), n = 0..9);
# Using the exponential generating functions of the columns:
EGFcol := proc(k, len) local egf, ser, n; egf := (1 - 2*x)^(-k/2);
ser := series(egf, x, len+2): seq(n!*coeff(ser, x, n), n = 0..len) end:
seq(lprint(EGFcol(n, 9)), n = 0..8);
# Using the generating polynomials for the rows:
P := (n, x) -> local k; add(Stirling1(n, k)*(-2)^(n - k)*x^k, k=0..n):
seq(lprint([n], seq(P(n, k), k = 0..8)), n = 0..5);
# Implementing the comment of Werner Schulte about the LU decomposition of A:
with(LinearAlgebra):
L := Matrix(7, 7, (n, k) -> A371025(n - 1,  k - 1)):
U := Matrix(7, 7, (n, k) -> binomial(n - 1, k - 1)):
MatrixMatrixMultiply(L, Transpose(U));  #  Peter Luschny, Mar 08 2024
  • Mathematica
    A370419[n_, k_] := 2^n*Pochhammer[k/2, n];
Table[A370419[n-k, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Mar 06 2024 *)
  • SageMath
    def A(n, k): return 2**n * rising_factorial(k/2, n)
for n in range(6): print([A(n, k) for k in range(9)])

Formula

The polynomials P(n, x) = Sum_{k=0..n} Stirling1(n, k)*(-2)^(n-k)*x^k are ordinary generating functions for row n, i.e., A(n, k) = P(n, k).
From Werner Schulte, Mar 07 2024: (Start)
A(n, k) = Product_{i=1..n} (2*i - 2 + k).
E.g.f. of column k: Sum_{n>=0} A(n, k) * t^n / (n!) = (1/sqrt(1 - 2*t))^k.
A(n, k) = A(n+1, k-2) / (k - 2) for k > 2.
A(n, k) = Sum_{i=0..k-1} i! * A265649(n, i) * binomial(k-1, i) for k > 0.
E.g.f. of row n > 0: Sum_{k>=1} A(n, k) * x^k / (k!) = (Sum_{k=1..n} A035342(n, k) * x^k) * exp(x).
Sum_{n>=0, k>=0} A(n, k) * x^k * t^n / (k! * n!) = exp(x/sqrt(1 - 2*t)).
Sum_{n>=0, k>=0} A(n, k) * x^k * t^n / (n!) = 1 / (1 - x/sqrt(1 - 2*t)).
The LU decomposition of this array is given by the upper triangular matrix U which is the transpose of A007318 and the lower triangular matrix L, where L is defined L(n, k) = A035342(n, k) * k! for 1 <= k <= n and L(n, 0) = 0^n. Note that L(n, k) + L(n, k+1) = A265649(n, k) * k! for 0 <= k <= n. (End)

A328489 Odd numbers k such that the four consecutive odd numbers starting with k have a total of 5 prime factors counting multiplicity.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 37, 67, 107, 307, 877, 1297, 2267, 2657, 3457, 3847, 3917, 4787, 4967, 5737, 11827, 12037, 14627, 16447, 18127, 18517, 19417, 20477, 20747, 20897, 21377, 21557, 22567, 22637, 23057, 23557, 23627, 25577, 29567, 31387, 32057, 33347, 33767, 34757, 35797, 36467, 36787, 37307
Offset: 1

Views

Author

J. M. Bergot and Robert Israel, Oct 16 2019

Keywords

Comments

Numbers k such that A001222(A190577(k))=5.
There are three cases:
k=3.
k, k+4 and k+6 are primes while k+2 is 3 times a prime.
k, k+2 and k+6 are primes while k+4 is 3 times a prime.
All terms > 13 have final digit 7.
The first n for which a(n+1)-a(n)=10 is 7538. - Robert Israel, Oct 19 2019

Examples

			a(3)=7 is in the sequence because 7*9*11*13 is the product of exactly 5 primes: 3*3*7*11*13.

Links

Crossrefs

Cf. A001222, A190577.

Programs

  • Maple
    A1:= select(t -> isprime((t+2)/3) and isprime(t) and isprime(t+4) and isprime(t+6), {seq(i,i=7..100000,30)}):
A2:= select(t -> isprime((t+4)/3) and isprime(t) and isprime(t+2) and isprime(t+6), {seq(i,i=17..100000,30)});
sort(convert({3,5,11,13} union A1 union A2,list));

A346514 a(n) = n^4 + 28*n^3 + 252*n^2 + 784*n + 448.

Original entry on oeis.org

448, 1513, 3264, 5905, 9664, 14793, 21568, 30289, 41280, 54889, 71488, 91473, 115264, 143305, 176064, 214033, 257728, 307689, 364480, 428689, 500928, 581833, 672064, 772305, 883264, 1005673, 1140288, 1287889, 1449280, 1625289, 1816768, 2024593, 2249664, 2492905, 2755264
Offset: 0

Views

Author

Lamine Ngom, Jul 21 2021

Keywords

Comments

The product of eight positive integers shifted by 2; i.e., m * (m+2) * (m+4) * ... * (m+14) = A346515(m) can always be expressed as the difference of two squares: x^2 - y^2.
This sequence gives the x-values for each product. The y-values are A152691(n+7).
More generally, for any k, we have n * (n+k) * (n+2*k) * ... * (n+7*k) = a(n,k) = x(n,k)^2 - y(n,k)^2, where
x(n,k) = n^4 + 14*k*n^3 + 63*k^2*n^2 + 98*k^3*n + 28*k^4,
y(n,k) = 4*k^3*(2*n + 7*k).
A239035(n) corresponds to a(n,k) in the case k = 1, with related y(n,k) = A346376(n).
This sequence is y(n,k) in the case k = 2, with related y(n,k) = A152691(n+7).

Links

Crossrefs

Cf. A152691, A239035, A190577, A346515.

Formula

a(n) = sqrt(A346515(n) + A152691(n+7)^2).
G.f.: (448 - 727*x + 179*x^2 + 235*x^3 - 111*x^4)/(1 - x)^5. - Stefano Spezia, Jul 22 2021

A346515 a(n) = n*(n+2)*(n+4)*(n+6)*(n+8)*(n+10)*(n+12)*(n+14).

Original entry on oeis.org

0, 2027025, 10321920, 34459425, 92897280, 218243025, 464486400, 916620705, 1703116800, 3011753745, 5109350400, 8365982625, 13284311040, 20534684625, 30996725760, 45808142625, 66421555200, 94670161425, 132843110400, 183771489825, 250925875200, 338526428625, 451666575360
Offset: 0

Views

Author

Lamine Ngom, Jul 21 2021

Keywords

Comments

a(n) can always be expressed as the difference of two squares: x^2 - y^2.
A346514(n) gives the x-values for each product. The y-values being A152691(n+7).
More generally, for any k, we have: n*(n+k)*(n+2*k)*...*(n+7*k) = a(n,k) = x(n,k)^2 - y(n,k)^2, where
x(n,k) = n^4 + 14*k*n^3 + 63*k^2*n^2 + 98*k^3*n + 28*k^4,
y(n,k) = 8*k^3*n + 28*k^4.
A239035(n) corresponds to a(n,k) in the case k = 1, with related y(n,k) = A346376(n).

Links

Crossrefs

Cf. A239035, A190577, A346514, A346376.

Programs

  • Mathematica
    a[n_] := (n + 14)!!/(n - 2)!!; Array[a, 23, 0] (* Amiram Eldar, Jul 22 2021 *)

Formula

a(n) = A346514(n)^2 - A152691(n+7)^2.

A370912 a(n) = n*(n + 2)*(n + 4).

Original entry on oeis.org

0, 15, 48, 105, 192, 315, 480, 693, 960, 1287, 1680, 2145, 2688, 3315, 4032, 4845, 5760, 6783, 7920, 9177, 10560, 12075, 13728, 15525, 17472, 19575, 21840, 24273, 26880, 29667, 32640, 35805, 39168, 42735, 46512, 50505, 54720, 59163, 63840, 68757, 73920
Offset: 0

Views

Author

Peter Luschny, Mar 05 2024

Keywords

Links

Crossrefs

Cases of A370419(n, k): A000012 (n=0), A001477 (n=1), A005563 (n=2), this sequence (n=3), A190577(n=4).
Cf. A077415, A370890.
Cf. A028347, A028387, A211441.

Programs

  • Maple
    a := n -> n*(n + 2)*(n + 4): seq(a(n), n = 0..40);
# Using the generating function:
gf := 3*x*(x^2 - 4*x + 5)/(x - 1)^4: ser := series(gf, x, 42):
seq(coeff(ser, x, n), n = 0..40);
  • Mathematica
    Table[n(n+2)(n+4), {n,0,40}] (* or *) CoefficientList[Series[3*x*(x^2 - 4*x + 5)/(x - 1)^4,{x,0,40}],x] (* James C. McMahon, Mar 05 2024 *)

Formula

a(n) = 8*Pochhammer(n/2, 3).
a(n) = [x^n] 3*x*(x^2 - 4*x + 5)/(x - 1)^4.
a(n) = 3 * A077415(n + 2).
From Klaus Purath, Aug 02 2024: (Start)
a(n)^2 = A028347(n+2)^3 + 4*A028347(n+2)^2.
a(n+1) - a(n) = A211441(n+2).
a(n) = 3*Sum_{i = 1..n} A028387(i). (End)
E.g.f.: exp(x)*x*(15 + 9*x + x^2). - Stefano Spezia, Aug 18 2024
From Amiram Eldar, Oct 03 2024: (Start)
Sum_{n>=1} 1/a(n) = 11/96.
Sum_{n>=1} (-1)^(n+1)/a(n) = 5/96. (End)
Showing 1-5 of 5 results.

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