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 A327916 #44 Jul 13 2023 08:17:41
%S A327916 1,4,3,12,8,5,32,20,12,7,80,48,28,16,9,192,112,64,36,20,11,448,256,
%T A327916 144,80,44,24,13,1024,576,320,176,96,52,28,15,2304,1280,704,384,208,
%U A327916 112,60,32,17,5120,2816,1536,832,448,240,128,68,36,19,11264,6144,3328,1792,960,512,272,144,76,40,21
%N A327916 Triangle T(k, n) read by rows: Array A(k, n) = 2^k*(k + 1 + 2*n), k >= 0, n >= 0, read by antidiagonals upwards.
%C A327916 The array A(k, n) arises from the following Pascal-type triangles PTodd(k), k >= 0 based on the positive odd integers A005408.
%C A327916 For example, the Pascal-type triangle PTodd(k), for k = 3 is
%C A327916 1 3 5 7
%C A327916 4 8 12
%C A327916 12 20
%C A327916 32
%C A327916 Taken upside-down such triangles become so-called addition towers of height k+1 (Rechenturm in German elementary schools; thanks to my correspondent Bennet D.), starting with any k+1 numbers. Here the positive odd numbers are used.
%C A327916 The sequence s of the final number of these Pascal-type triangles PT(k), for k >= 0, begins 1, 4, 12, 32, ...; s(k) = (k+1)*2^k = A001787(k+1), for k >= 0.
%C A327916 For k -> infinity the left-aligned row sequences build the array A(k, n), with k >= 0 and n >= 0, namely A(k, n) = 2^k*(k + 2*n + 1); this array begins:
%C A327916 k\n 0 1 2 3 4 5 ...
%C A327916 -------------------------------
%C A327916 0: 1 3 5 7 9 11 ... {A005408(n)}
%C A327916 1: 4 8 12 16 20 24 ... {A008586(n+1)}
%C A327916 2: 12 20 28 36 44 52 ... {A017113(n+1)}
%C A327916 3: 32 48 64 80 96 112 ... {A008598(n+2)}
%C A327916 4: 80 112 144 176 208 240 ... {16*A005408(n+2)}
%C A327916 5: 192 256 320 384 448 512 ... {A152691(n+3)}
%C A327916 6: 448 576 704 832 960 1088 ... {64*A005408(n+3)}
%C A327916 ...
%C A327916 The sequence s, the first (n=0) column of A, is always the binomial transform of the first (k=0) row in A.
%C A327916 A(k, n) = Sum_{j=0..k} binomial(k, j)*(2*(n+j)+1) = 2^k*(k + 1 + 2*n), for k >= 0 and n >= 0.
%C A327916 The corresponding antidiagonal-upwards read triangle is T(k, n) = A(k-n, n) = 2^(k-n)*(k + n + 1), n >= 0, k = 0..n.
%C A327916 If the nonnegative integers A001477 are used as k = 0 row of the array Anneg(k, n) = 2^(k-1)*(2*n + k), for k >= 0, n >= 0, with the triangle Tnneg(k, n) = Anneg(k-n, n) = (n + k)*2^(k-n-1), k >= 0, n = 0..k, then the s sequence is snneg(k) = Tnneg(k, 0) = k*2^{k-1} = A001787(k), the binomial transform of the sequence{A001477(n)}_{n>=0}. The triangle Tnneg begins [0], [1, 1], [4, 3, 2], [12, 8, 5, 3], [32, 20, 12, 7, 4], ... . See A062111 and the row-reversed triangle A152920 for other versions.
%F A327916 Array A(k, n) = Sum_{j=0..k} binomial(k, j)*(2*(n+j) + 1) = 2^k*(k + 1+ 2*n), for k >= 0 and n >= 0.
%F A327916 Triangle T(k, n) = A(k-n, n) = 2^(k-n)*(k + n + 1), n >= 0, k = 0..n.
%F A327916 Recurrence: T(k, 0) = (k+1)*2^k = A001787(k+1), for k >= 0, and T(k, n) = T(k, n-1) - T(k-1, n-1), for n >= 1, k >= 1, with T(k, n) = 0 if k < n.
%F A327916 O.g.f. for row polynomials: G(z,x) = Sum_{n=0..k} R(k, x)*z^n =
%F A327916 (1 + x*z*(1 - 4*z))/((1 - 2*z)^2*(1 - x*z)^2).
%F A327916 T(k, 0) = Sum_{n=0..k} binomial(k,n)*T(n, n), k >= 0 (binomial transform).
%e A327916 The triangle T(k, n) begins:
%e A327916 k\n 0 1 2 3 4 5 6 7 8 9 10 ...
%e A327916 -----------------------------------------------------
%e A327916 0: 1
%e A327916 1: 4 3
%e A327916 2: 12 8 5
%e A327916 3: 32 20 12 7
%e A327916 4: 80 48 28 16 9
%e A327916 5: 192 112 64 36 20 11
%e A327916 6: 448 256 144 80 44 24 13
%e A327916 7: 1024 576 320 176 96 52 28 15
%e A327916 8: 2304 1280 704 384 208 112 60 32 17
%e A327916 9: 5120 2816 1536 832 448 240 128 68 36 19
%e A327916 10: 11264 6144 3328 1792 960 512 272 144 76 40 21
%e A327916 ...
%t A327916 Table[2^#*(# + 1 + 2 n) &[k - n], {k, 0, 10}, {n, 0, k}] // Flatten (* _Michael De Vlieger_, Oct 03 2019 *)
%Y A327916 Column sequences without leading zeros are for n=0..9: A001787(n+1), A001792(n+1), A045623(n+2), A045891(n+3), A034007(n+4), A111297(n+3), A159694(n+1), A159695(n+1), A159696(n+1), A159697(n+1).
%Y A327916 The sequence of (sub)diagonal k, for k >= 0, is the row k sequence of array A: {(k + 2*n + 1)*2^k}_{k >= 0}.
%Y A327916 Row sums: A213569(k+1), k >= 0 (see the _J. M. Bergot_ comments there).
%Y A327916 Cf. A006211, A152920.
%K A327916 nonn,easy,tabl
%O A327916 0,2
%A A327916 _Wolfdieter Lang_, Oct 03 2019
%E A327916 Definition corrected by _Georg Fischer_, Jul 13 2023