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.
%I A371077 #28 Aug 17 2025 02:07:34
%S A371077 1,0,1,0,1,1,0,4,2,1,0,28,10,3,1,0,280,80,18,4,1,0,3640,880,162,28,5,
%T A371077 1,0,58240,12320,1944,280,40,6,1,0,1106560,209440,29160,3640,440,54,7,
%U A371077 1,0,24344320,4188800,524880,58240,6160,648,70,8,1
%N A371077 Square array read by ascending antidiagonals: A(n, k) = 3^n*Pochhammer(k/3, n).
%H A371077 Paolo Xausa, <a href="/A371077/b371077.txt">Table of n, a(n) for n = 0..11324</a> (first 150 antidiagonals, flattened).
%F A371077 A(n, k) = Product_{j=0..n-1} (3*j + k).
%F A371077 A(n, k) = A(n+1, k-3) / (k - 3) for k > 3.
%F A371077 A(n, k) = Sum_{j=0..n} Stirling1(n, j)*(-3)^(n - j)* k^j.
%F A371077 A(n, k) = k! * [x^k] (exp(x) * p(n, x)), where p(n, x) are the row polynomials of A371080.
%F A371077 E.g.f. of column k: (1 - 3*t)^(-k/3).
%F A371077 E.g.f. of row n: exp(x) * (Sum_{k=0..n} A371076(n, k) * x^k / (k!)).
%F A371077 Sum_{n>=0, k>=0} A(n, k) * x^k * t^n / (n!) = 1/(1 - x/(1 - 3*t)^(1/3)).
%F A371077 Sum_{n>=0, k>=0} A(n, k) * x^k * t^n /(n! * k!) = exp(x/(1 - 3*t)^(1/3)).
%F A371077 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 = A371076, i.e., A(n, k) = Sum_{i=0..k} A371076(n, i) * binomial(k, i).
%e A371077 The array starts:
%e A371077 [0] 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e A371077 [1] 0, 1, 2, 3, 4, 5, 6, 7, 8, ...
%e A371077 [2] 0, 4, 10, 18, 28, 40, 54, 70, 88, ...
%e A371077 [3] 0, 28, 80, 162, 280, 440, 648, 910, 1232, ...
%e A371077 [4] 0, 280, 880, 1944, 3640, 6160, 9720, 14560, 20944, ...
%e A371077 [5] 0, 3640, 12320, 29160, 58240, 104720, 174960, 276640, 418880, ...
%e A371077 .
%e A371077 Seen as the triangle T(n, k) = A(n - k, k):
%e A371077 [0] 1;
%e A371077 [1] 0, 1;
%e A371077 [2] 0, 1, 1;
%e A371077 [3] 0, 4, 2, 1;
%e A371077 [4] 0, 28, 10, 3, 1;
%e A371077 [5] 0, 280, 80, 18, 4, 1;
%e A371077 [6] 0, 3640, 880, 162, 28, 5, 1;
%e A371077 [7] 0, 58240, 12320, 1944, 280, 40, 6, 1;
%e A371077 [8] 0, 1106560, 209440, 29160, 3640, 440, 54, 7, 1;
%e A371077 .
%e A371077 Illustrating the LU decomposition of A:
%e A371077 / 1 \ / 1 1 1 1 1 ... \ / 1 1 1 1 1 ... \
%e A371077 | 0 1 | | 1 2 3 4 ... | | 0 1 2 3 4 ... |
%e A371077 | 0 4 2 | * | 1 3 6 ... | = | 0 4 10 18 28 ... |
%e A371077 | 0 28 24 6 | | 1 4 ... | | 0 28 80 162 280 ... |
%e A371077 | 0 280 320 144 24 | | 1 ... | | 0 280 880 1944 3640 ... |
%e A371077 | . . . | | . . . | | . . . |
%p A371077 A := (n, k) -> 3^n*pochhammer(k/3, n):
%p A371077 A := (n, k) -> local j; mul(3*j + k, j = 0..n-1):
%p A371077 # Read by antidiagonals:
%p A371077 T := (n, k) -> A(n - k, k): seq(seq(T(n, k), k = 0..n), n = 0..9);
%p A371077 seq(lprint([n], seq(T(n, k), k = 0..n)), n = 0..9);
%p A371077 # Using the generating polynomials of the rows:
%p A371077 P := (n, x) -> local k; add(Stirling1(n, k)*(-3)^(n - k)*x^k, k=0..n):
%p A371077 seq(lprint([n], seq(P(n, k), k = 0..9)), n = 0..5);
%p A371077 # Using the exponential generating functions of the columns:
%p A371077 EGFcol := proc(k, len) local egf, ser, n; egf := (1 - 3*x)^(-k/3);
%p A371077 ser := series(egf, x, len+2): seq(n!*coeff(ser, x, n), n = 0..len) end:
%p A371077 seq(lprint([k], EGFcol(k, 8)), k = 0..6);
%p A371077 # As a matrix product:
%p A371077 with(LinearAlgebra):
%p A371077 L := Matrix(7, 7, (n, k) -> A371076(n - 1, k - 1)):
%p A371077 U := Matrix(7, 7, (n, k) -> binomial(n - 1, k - 1)):
%p A371077 MatrixMatrixMultiply(L, Transpose(U));
%t A371077 Table[3^(n-k)*Pochhammer[k/3, n-k], {n, 0, 10}, {k, 0, n}] (* _Paolo Xausa_, Mar 14 2024 *)
%o A371077 (SageMath)
%o A371077 def A(n, k): return 3**n * rising_factorial(k/3, n)
%o A371077 def A(n, k): return (-3)**n * falling_factorial(-k/3, n)
%Y A371077 Family m^n*Pochhammer(k/m, n): A094587 (m=1), A370419 (m=2), this sequence (m=3), A370915 (m=4).
%Y A371077 Rows: A000012, A001477, A028552.
%Y A371077 Columns: A000007, A007559, A008544, A032031, A007559 (shifted), A034000, A034001, A051604, A051605, A051606, A051607, A051608, A051609.
%Y A371077 Cf. A303486 (main diagonal), A371079 (row sums of triangle), A371076, A371080.
%K A371077 nonn,tabl,easy
%O A371077 0,8
%A A371077 _Werner Schulte_ and _Peter Luschny_, Mar 10 2024