Real-rooted atlas

A007318 → −D²+(−1+n)D+1+n → x ↦ x+1 → (θ+1)²

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(8 + 4\,x - 20\,x^{2} + 2\,x^{3} + 4\,n + 14\,n x + 38\,n x^{2} - 9\,n x^{3} + 4\,n^{2} + 8\,n^{2} x - 2\,n^{2} x^{2} + 15\,n^{2} x^{3} + 6\,n^{3} x + 7\,n^{3} x^{2} - n^{3} x^{3} + n^{3} x^{4} + n^{4} x^{2} + n^{4} x^{3}\right)\,P_{n}(x) = \left(32 + 80\,x + 80\,x^{2} + 40\,x^{3} + 10\,x^{4} + x^{5} + 24\,n + 108\,n x + 166\,n x^{2} + 103\,n x^{3} + 28\,n x^{4} + 3\,n x^{5} + 8\,n^{2} + 56\,n^{2} x + 76\,n^{2} x^{2} + 61\,n^{2} x^{3} + 24\,n^{2} x^{4} + 3\,n^{2} x^{5} + 12\,n^{3} x + 28\,n^{3} x^{2} + 17\,n^{3} x^{3} + 5\,n^{3} x^{4} + n^{3} x^{5} + 2\,n^{4} x^{2} + 3\,n^{4} x^{3} + n^{4} x^{4}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
14812
214562797
34426427064642
4128102416208321253729
5352352075606080200021619728
692811136302403328017250410434397281
7236833152108864152320108500408247546512454086
858889420836288061644856000029030485064128007292028321
91433625804811404802279424252000016692486750241612802041210008739252
1034304686080342144078643201024800082736644280640141312028431031000133136538209
1180896177971298841602568192038544000367234562318131297996802726460473000452541728148921578
121884164521984276756488019968013622400014963097611156006457581568205286404928000750684639362197593855793
13434176112885767547904024131993645760000056932761648970521629872128013010025640040000844386411456648788027442323696248
149912322775449620127744070372556814734720002047997952199649049614057472007238911682722720007364156813837824169169011799233758942912001
Technical details
Node id
n-fb8d40c1cf430e80 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-1f33a1ea2883bcf1

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:     1;
  1:     4,      8;
  2:    14,     56,      27;
  3:    44,    264,     270,      64;
  4:   128,   1024,    1620,     832,     125;
  5:   352,   3520,    7560,    6080,    2000,     216;
  6:   928,  11136,   30240,   33280,   17250,    4104,    343;
  7:  2368,  33152,  108864,  152320,  108500,   40824,   7546,    512;
  8:  5888,  94208,  362880,  616448,  560000,  290304,  85064,  12800,   729;
  9: 14336, 258048, 1140480, 2279424, 2520000, 1669248, 675024, 161280, 20412, 1000;
%F Formula
The row polynomials satisfy (4*n^2+4*n+8+(6*n^3+8*n^2+14*n+4)*x+(n^4+7*n^3-2*n^2+38*n-20)*x^2+(n^4-n^3+15*n^2-9*n+2)*x^3+n^3*x^4)*P(n,x) = (8*n^2+24*n+32+(12*n^3+56*n^2+108*n+80)*x+(2*n^4+28*n^3+76*n^2+166*n+80)*x^2+(3*n^4+17*n^3+61*n^2+103*n+40)*x^3+(n^4+5*n^3+24*n^2+28*n+10)*x^4+(n^3+3*n^2+3*n+1)*x^5)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^2 + 24*n + 32 + (12*n^3 + 56*n^2 + 108*n + 80)*x + (2*n^4 + 28*n^3 + 76*n^2 + 166*n + 80)*x^2 + (3*n^4 + 17*n^3 + 61*n^2 + 103*n + 40)*x^3 + (n^4 + 5*n^3 + 24*n^2 + 28*n + 10)*x^4 + (n^3 + 3*n^2 + 3*n + 1)*x^5)*P[n - 1, x])/(4*n^2 + 4*n + 8 + (6*n^3 + 8*n^2 + 14*n + 4)*x + (n^4 + 7*n^3 - 2*n^2 + 38*n - 20)*x^2 + (n^4 - n^3 + 15*n^2 - 9*n + 2)*x^3 + n^3*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((8*n^2 + 24*n + 32 + (12*n^3 + 56*n^2 + 108*n + 80)*x + (2*n^4 + 28*n^3 + 76*n^2 + 166*n + 80)*x^2 + (3*n^4 + 17*n^3 + 61*n^2 + 103*n + 40)*x^3 + (n^4 + 5*n^3 + 24*n^2 + 28*n + 10)*x^4 + (n^3 + 3*n^2 + 3*n + 1)*x^5)*v[i - 1])/(4*n^2 + 4*n + 8 + (6*n^3 + 8*n^2 + 14*n + 4)*x + (n^4 + 7*n^3 - 2*n^2 + 38*n - 20)*x^2 + (n^4 - n^3 + 15*n^2 - 9*n + 2)*x^3 + n^3*x^4)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.