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.
Triangle begins as: 3; 3, 12; 3, 15, 105; 3, 18, 162, 1944; 3, 21, 231, 3465, 65835; 3, 24, 312, 5616, 129168, 3616704; 3, 27, 405, 8505, 229635, 7577955, 295540245; 3, 30, 510, 12240, 379440, 14418720, 648842400, 33739804800;
m:=2; function T(n,k) if k eq 0 then return NthPrime(m); else return (&*[j*n + NthPrime(m): j in [0..k]]); end if; return T; end function; [T(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 03 2019
m:=2; seq(seq(`if`(k=0, ithprime(m), mul(j*n + ithprime(m), j=0..k)), k=0..n), n=0..10); # G. C. Greubel, Dec 03 2019
T[n_, k_, m_]:= If[k==0, Prime[m], Product[j*n + Prime[m], {j,0,k}]];
Table[T[n,k,2], {n,0,10}, {k,0,n}]//Flatten
T(n,k) = my(m=2); if(k==0, prime(m), prod(j=0,k, j*n + prime(m)) ); \\ G. C. Greubel, Dec 03 2019
def T(n, k):
m=2
if (k==0): return nth_prime(m)
else: return product(j*n + nth_prime(m) for j in (0..k))
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Dec 03 2019
Array starts: [1] 1, 0, 0, 0, 0, 0, 0, 0, ... A000007 [2] 1, 1, 3, 15, 105, 945, 10395, 135135, ... A001147 [3] 1, 2, 10, 80, 880, 12320, 209440, 4188800, ... A008544 [4] 1, 3, 21, 231, 3465, 65835, 1514205, 40883535, ... A008545 [5] 1, 4, 36, 504, 9576, 229824, 6664896, 226606464, ... A008546 [6] 1, 5, 55, 935, 21505, 623645, 21827575, 894930575, ... A008543 [7] 1, 6, 78, 1560, 42120, 1432080, 58715280, 2818333440, ... A049209 [8] 1, 7, 105, 2415, 74865, 2919735, 137227545, 7547514975, ... A049210 [9] 1, 8, 136, 3536, 123760, 5445440, 288608320, 17893715840, ... A049211 Triangle starts: [1] [1] [2] [1, 0] [3] [1, 1, 0] [4] [1, 2, 3, 0] [5] [1, 3, 10, 15, 0] [6] [1, 4, 21, 80, 105, 0] [7] [1, 5, 36, 231, 880, 945, 0] [8] [1, 6, 55, 504, 3465, 12320, 10395, 0] [9] [1, 7, 78, 935, 9576, 65835, 209440, 135135, 0]
[k eq n select 0^(n-1) else Round((n-k+1)^(k-1)*Gamma(k-1 + (n-k)/(n-k+1))/Gamma((n-k)/(n-k+1))): k in [1..n], n in [1..10]]; // G. C. Greubel, Feb 22 2022
A[n_, k_] := -(-n)^k * FactorialPower[1/n, k]; Table[A[n - k + 1, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Amiram Eldar, Dec 21 2021 *)
def A(n, k): return -(-n)^k*falling_factorial(1/n, k) def T(n, k): return A(n-k+1, k) for n in (1..9): print([A(n, k) for k in (1..8)]) for n in (1..9): print([T(n, k) for k in (1..n)])
(1 + x)^(1/4) = 1 + x/4 - 3*x^2/32 + 7*x^3/128 - 77*x^4/2048 + 231*x^5/8192 - 1463*x^6/65536 + ... Coefficients are 1, 1/4, -3/32, 7/128, -77/2048, 231/8192, -1463/65536, ...
nmax = 24; CoefficientList[Series[(1 + x)^(1/4), {x, 0, nmax}], x] // Numerator
Table[Binomial[1/4, n], {n, 0, 24}] // Numerator
my(x='x+O('x^30)); apply(numerator, Vec((1 + x)^(1/4))) \\ Michel Marcus, Aug 02 2023
[n\k][ 0, 1, 2, 3, 4, 5, 6 ] [0] 1, [1] 3, 4, [2] 21, 40, 16, [3] 231, 524, 336, 64, [4] 3465, 8784, 7136, 2304, 256, [5] 65835, 180756, 170720, 72320, 14080, 1024, [6] 1514205, 4420728, 4649584, 2346240, 613120, 79872, 4096.
s[][0, 0] = 1; s[m][n_, k_] /; (k > n || k < 0) = 0; s[m_][n_, k_] := s[m][n, k] = s[m][n - 1, k - 1] + (m*n - 1)*s[m][n - 1, k]; T[n_, k_] := 4^k*s[4][n, k]; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 02 2019 *)
@CachedFunction
def SF_CS(n, k, m):
if k > n or k < 0 : return 0
if n == 0 and k == 0: return 1
return m*SF_CS(n-1, k-1, m) + (m*n-1)*SF_CS(n-1, k, m)
for n in (0..8): [SF_CS(n, k, 4) for k in (0..n)]
a(2) = 131 because sigma[4,3]^{(3)}_2 = 3*(7 + 11) + 7*11 = 131. There are three rectangles (2D rectangular polytopes) built from two orthogonal vectors of different lengths from the set of {3,7,11} of total area 131.
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