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 A030528 #125 Jun 27 2025 23:17:24
%S A030528 1,1,1,0,2,1,0,1,3,1,0,0,3,4,1,0,0,1,6,5,1,0,0,0,4,10,6,1,0,0,0,1,10,
%T A030528 15,7,1,0,0,0,0,5,20,21,8,1,0,0,0,0,1,15,35,28,9,1,0,0,0,0,0,6,35,56,
%U A030528 36,10,1,0,0,0,0,0,1,21,70,84,45,11,1,0,0,0,0,0,0,7,56,126,120,55,12,1
%N A030528 Triangle read by rows: a(n,k) = binomial(k,n-k).
%C A030528 A convolution triangle of numbers obtained from A019590.
%C A030528 a(n,m) := s1(-1; n,m), a member of a sequence of triangles including s1(0; n,m)= A023531(n,m) (unit matrix) and s1(2; n,m)= A007318(n-1,m-1) (Pascal's triangle).
%C A030528 The signed triangular matrix a(n,m)*(-1)^(n-m) is the inverse matrix of the triangular Catalan convolution matrix A033184(n+1,m+1), n >= m >= 0, with A033184(n,m) := 0 if n<m.
%C A030528 Riordan array (1+x, x(1+x)). The signed triangle is the Riordan array (1-x,x(1-x)), inverse to (c(x),xc(x)) with c(x) g.f. for A000108. - _Paul Barry_, Feb 02 2005 [with offset 0]
%C A030528 Also, a(n,k)=number of compositions of n into k parts of 1's and 2's. Example: a(6,4)=6 because we have 2211, 2121, 2112, 1221, 1212 and 1122. - _Emeric Deutsch_, Apr 05 2005 [see MacMahon and Riordan. - _Wolfdieter Lang_, Jul 27 2023]
%C A030528 Subtriangle of A026729. - _Philippe Deléham_, Aug 31 2006
%C A030528 a(n,k) is the number of length n-1 binary sequences having no two consecutive 0's with exactly k-1 1's. Example: a(6,4)=6 because we have 01011, 01101, 01110, 10101, 10110, 11010. - _Geoffrey Critzer_, Jul 22 2013
%C A030528 Mirrored, shifted Fibonacci polynomials of A011973. The polynomials (illustrated below) of this entry have the property that p(n,t) = t * [p(n-1,t) + p(n-2,t)]. The additive properties of Pascal's triangle (A007318) are reflected in those of these polynomials, as can be seen in the Example Section below and also when the o.g.f. G(x,t) below is expanded as the series x*(1+x) + t * [x*(1+x)]^2 + t^2 * [x*(1+x)]^3 + ... . See also A053122 for a relation to Cartan matrices. - _Tom Copeland_, Nov 04 2014
%C A030528 Rows of this entry appear as columns of an array for an infinitesimal generator presented in the Copeland link. - _Tom Copeland_, Dec 23 2015
%C A030528 For n >= 2, the n-th row is also the coefficients of the vertex cover polynomial of the (n-1)-path graph P_{n-1}. - _Eric W. Weisstein_, Apr 10 2017
%C A030528 With an additional initial matrix element a_(0,0) = 1 and column of zeros a_(n,0) = 0 for n > 0, these are antidiagonals read from bottom to top of the numerical coefficients of the Maurer-Cartan form matrix of the Leibniz group L^(n)(1,1) presented on p. 9 of the Olver paper, which is generated as exp[c. * M] with (c.)^n = c_n and M the Lie infinitesimal generator A218272. Cf. A011973. And A169803. - _Tom Copeland_, Jul 02 2018
%D A030528 P. A. MacMahon, Combinatory Analysis, Two volumes (bound as one), Chelsea Publishing Company, New York, 1960, Vol. I, nr. 124, p. 151.
%D A030528 John Riordan, An Introduction to Combinatorial Analysis, John Wiley & Sons, London, 1958. eq. (35), p.124, 11. p. 154.
%H A030528 Indranil Ghosh, <a href="/A030528/b030528.txt">Rows 1..125 of triangle, flattened</a>
%H A030528 Jean Luc Baril, Rigoberto Flórez, and José L. Ramirez, <a href="http://jl.baril.u-bourgogne.fr/narayana.pdf">Generalized Narayana arrays, restricted Dyck paths, and related bijections</a>, Univ. Bourgogne (France, 2025). See p. 18.
%H A030528 Tom Copeland, <a href="http://tcjpn.wordpress.com/2015/12/21/generators-inversion-and-matrix-binomial-and-integral-transforms/">Generators, Inversion, and Matrix, Binomial, and Integral Transforms</a>, <a href="http://tcjpn.wordpress.com/2015/10/12/the-elliptic-lie-triad-kdv-and-ricattt-equations-infinigens-and-elliptic-genera/">Addendum to Elliptic Lie Triad</a>
%H A030528 Loïc Foissy, <a href="https://arxiv.org/abs/2501.18212">Cointeraction on noncrossing partitions and related polynomial invariants</a>, arXiv:2501.18212 [math.CO], 2025. See p. 21.
%H A030528 Wolfdieter Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/LANG/lang.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%H A030528 Donatella Merlini, Renzo Sprugnoli, and M. Cecilia Verri, <a href="http://local.disia.unifi.it/merlini/papers/Lucidi.ps">An algebra for proper generating trees</a>, Colloquium on Mathematics and Computer Science, Versailles, September 2000.
%H A030528 Donatella Merlini, Renzo Sprugnoli, and M. Cecilia Verri, <a href="http://dx.doi.org/10.1007/978-3-0348-8405-1_11">An algebra for proper generating trees</a>, Mathematics and Computer Science, Part of the series Trends in Mathematics pp 127-139.
%H A030528 Peter J. Olver, <a href="https://www-users.cse.umn.edu/~olver/di_/contact.pdf">The canonical contact form</a>.
%H A030528 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PathGraph.html">Path Graph</a>
%H A030528 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/VertexCoverPolynomial.html">Vertex Cover Polynomial</a>
%F A030528 a(n, m) = 2*(2*m-n+1)*a(n-1, m)/n + m*a(n-1, m-1)/n, n >= m >= 1; a(n, m) := 0, n<m; a(n, 0) := 0; a(1, 1)=1.
%F A030528 G.f. for m-th column: (x*(1+x))^m.
%F A030528 As a number triangle with offset 0, this is T(n, k) = Sum_{i=0..n} (-1)^(n+i)*binomial(n, i)*binomial(i+k+1, 2k+1). The antidiagonal sums give the Padovan sequence A000931(n+5). Inverse binomial transform of A078812 (product of lower triangular matrices). - _Paul Barry_, Jun 21 2004
%F A030528 G.f.: (1 + x)/(1 - y*x - y*x^2). - _Geoffrey Critzer_, Jul 22 2013 [offset 0] [with offset 1: g.f. of row polynomials in y: x*(1+x)*y/(1 - x*(1+x)*y). - _Wolfdieter Lang_, Jul 27 2023]
%F A030528 From _Tom Copeland_, Nov 04 2014: (Start)
%F A030528 O.g.f: G(x,t) = x*(1+x) / [1 - t*x*(1+x)] = -P[Cinv(-x),t], where P(x,t)= x / (1 + t*x) and Cinv(x)= x*(1-x) are the compositional inverses in x of Pinv(x,t) = -P(-x,t) = x / (1 - t*x) and C(x) = [1-sqrt(1-4*x)]/2, an o.g.f. for the shifted Catalan numbers A000108.
%F A030528 Therefore, Ginv(x,t) = -C[Pinv(-x,t)] = {-1 + sqrt[1 + 4*x/(1+t*x)]}/2, which is -A124644(-x,t).
%F A030528 This places this array in a family of arrays related by composition of P and C and their inverses and interpolation by t, such as A091867 and A104597, and associated to the Catalan, Motzkin, Fine, and Fibonacci numbers. Cf. A104597 (polynomials shifted in t) A125145, A146559, A057078, A000045, A155020, A125145, A039717, A001792, A057862, A011973, A115139. (End)
%e A030528 Triangle starts:
%e A030528 [ 1] 1
%e A030528 [ 2] 1 1
%e A030528 [ 3] 0 2 1
%e A030528 [ 4] 0 1 3 1
%e A030528 [ 5] 0 0 3 4 1
%e A030528 [ 6] 0 0 1 6 5 1
%e A030528 [ 7] 0 0 0 4 10 6 1
%e A030528 [ 8] 0 0 0 1 10 15 7 1
%e A030528 [ 9] 0 0 0 0 5 20 21 8 1
%e A030528 [10] 0 0 0 0 1 15 35 28 9 1
%e A030528 [11] 0 0 0 0 0 6 35 56 36 10 1
%e A030528 [12] 0 0 0 0 0 1 21 70 84 45 11 1
%e A030528 [13] 0 0 0 0 0 0 7 56 126 120 55 12 1
%e A030528 ...
%e A030528 From _Tom Copeland_, Nov 04 2014: (Start)
%e A030528 For quick comparison to other polynomials:
%e A030528 p(1,t) = 1
%e A030528 p(2,t) = 1 + 1 t
%e A030528 p(3,t) = 0 + 2 t + 1 t^2
%e A030528 p(4,t) = 0 + 1 t + 3 t^2 + 1 t^3
%e A030528 p(5,t) = 0 + 0 + 3 t^2 + 4 t^3 + 1 t^4
%e A030528 p(6,t) = 0 + 0 + 1 t^2 + 6 t^3 + 5 t^4 + 1 t^5
%e A030528 p(7,t) = 0 + 0 + 0 + 4 t^3 + 10 t^4 + 6 t^5 + 1 t^6
%e A030528 p(8,t) = 0 + 0 + 0 + 1 t^3 + 10 t^4 + 15 t^5 + 7 t^6 + 1 t^7
%e A030528 ...
%e A030528 Reading along columns gives rows for Pascal's triangle. (End)
%p A030528 for n from 1 to 12 do seq(binomial(k,n-k),k=1..n) od; # yields sequence in triangular form - _Emeric Deutsch_, Apr 05 2005
%t A030528 nn=10;CoefficientList[Series[(1+x)/(1-y x - y x^2),{x,0,nn}],{x,y}]//Grid (* _Geoffrey Critzer_, Jul 22 2013 *)
%t A030528 Table[Binomial[k, n - k], {n, 13}, {k, n}] // Flatten (* _Michael De Vlieger_, Dec 23 2015 *)
%t A030528 CoefficientList[Table[x^(n/2 - 1) Fibonacci[n + 1, Sqrt[x]], {n, 10}],
%t A030528 x] // Flatten (* _Eric W. Weisstein_, Apr 10 2017 *)
%o A030528 (Magma) /* As triangle */ [[Binomial(k, n-k): k in [1..n]]: n in [1.. 15]]; // _Vincenzo Librandi_, Nov 05 2014
%Y A030528 Row sums A000045(n+1) (Fibonacci). a(n, 1)= A019590(n) (Fermat's last theorem). Cf. A049403.
%Y A030528 Cf. A104597, A146559, A155020, A125145, A057078, A039717, A001792, A057862, A011973, A115139.
%K A030528 easy,nonn,tabl
%O A030528 1,5
%A A030528 _Wolfdieter Lang_
%E A030528 More terms from _Emeric Deutsch_, Apr 05 2005