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 A329154 #35 May 04 2025 09:02:42
%S A329154 1,1,1,2,2,1,6,6,3,1,26,24,12,4,1,158,130,60,20,5,1,1330,948,390,120,
%T A329154 30,6,1,15414,9310,3318,910,210,42,7,1,245578,123312,37240,8848,1820,
%U A329154 336,56,8,1,5382862,2210202,554904,111720,19908,3276,504,72,9,1
%N A329154 Coefficients of polynomials related to the sum of Gaussian binomial coefficients for q = 2. Triangle read by rows, T(n,k) for 0 <= k <= n.
%C A329154 T(n,k) is the number of n X n matrices over F_2 in reduced row echelon form having exactly k zero-columns. Equivalently, T(n,k) is the number of subspaces of F_2^n that "involve" n-k coordinates. (For the definition of "involve" see the link below: D. E. Knuth, Letter to Daniel Ullman and others). - _Geoffrey Critzer_, May 03 2025
%H A329154 D. E. Knuth, <a href="/A323841/a323841.pdf">Letter to Daniel Ullman and others</a>, Apr 29 1997 [Annotated scanned copy, with permission]
%H A329154 R. P. Stanley and S. C. Locke, <a href="https://www.jstor.org/stable/2589063">Graphs without increasing paths: Solution to Problem 10572</a>, The American Mathematical Monthly, 106(2) (1999), 168.
%F A329154 Let P(n, k, x) = x*P(n, k-1, x) + 2^k*P(n-1, k, (x+1)/2) and Q(n, x) = Sum_{k=0..n} P(n-k, k, x) then T(n, k) = [x^k] Q(n, x).
%F A329154 T(n, k) = (1/k!) * Pochhammer(n-k+1, k) * Sum_{j=0..n-k}((-1)^j*Sum_{m=0..n-k-j} (Product_{i=1..n-k-m-j} ((2^(i+m)-1)/(2^i-1))) * binomial(n-k, j)). - _Detlef Meya_, Oct 07 2023
%F A329154 T(n,k) = binomial(n,k)*A135922(n-k). (see Stanley-Locke link above) - _Geoffrey Critzer_, May 03 2025
%F A329154 E.g.f.: exp(y x)*f(x) where f(x) is the e.g.f. for A135922. - _Geoffrey Critzer_, May 03 2025
%e A329154 Triangle starts:
%e A329154 [0] [1]
%e A329154 [1] [1, 1]
%e A329154 [2] [2, 2, 1]
%e A329154 [3] [6, 6, 3, 1]
%e A329154 [4] [26, 24, 12, 4, 1]
%e A329154 [5] [158, 130, 60, 20, 5, 1]
%e A329154 [6] [1330, 948, 390, 120, 30, 6, 1]
%e A329154 [7] [15414, 9310, 3318, 910, 210, 42, 7, 1]
%e A329154 [8] [245578, 123312, 37240, 8848, 1820, 336, 56, 8, 1]
%e A329154 [9] [5382862, 2210202, 554904, 111720, 19908, 3276, 504, 72, 9, 1]
%p A329154 T := (n, k) -> local j, m; pochhammer(n - k + 1, k)*add((-1)^j*add(product((2^(i + m) - 1)/(2^i - 1), i = 1..n-k-m-j), m = 0..n-k-j)*binomial(n - k, j), j = 0..n-k) / k!: for n from 0 to 9 do seq(T(n,k), k=0..n) od; # _Peter Luschny_, Oct 08 2023
%t A329154 T[n_,k_]:= (Pochhammer[n-k+1,k]/(k!)*Sum[(-1)^j*Sum[Product[(2^(i+m)-1)/(2^i-1),{i,1,n-k-m-j}],{m,0,n-k-j}]*Binomial[n-k,j],{j,0,n-k}]); Flatten[Table[T[n,k],{n,0,9},{k,0,n}]] (* _Detlef Meya_, Oct 07 2023 *)
%o A329154 (Sage)
%o A329154 R = PolynomialRing(QQ, 'x')
%o A329154 x = R.gen()
%o A329154 @cached_function
%o A329154 def P(n, k, x):
%o A329154 if k < 0 or n < 0: return R(0)
%o A329154 if k == 0: return R(1)
%o A329154 return x*P(n, k-1, x) + 2^k*P(n-1, k, (x+1)/2)
%o A329154 def row(n): return sum(P(n-k, k, x) for k in range(n+1)).coefficients()
%o A329154 print(flatten([row(n) for n in range(10)]))
%Y A329154 Row sums: A006116, first column: A135922.
%Y A329154 Cf. A022166, A323842, A323843.
%K A329154 nonn,tabl
%O A329154 0,4
%A A329154 _Vladimir Kruchinin_ and _Peter Luschny_, Mar 08 2020