Real-rooted atlas

A135065 → x ↦ x+1 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(n^{2}\right)\,P_{n}(x) = \left(3 + x + 6\,n + 2\,n x + 3\,n^{2} + n^{2} x\right)\,P_{n-1}(x) + \left(x + 2\,n x + n^{2} x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
112416
281639153
3432592192161232
4202543752425450259300
58748281162376073809003668940
6357211649831999699261018620161749513569
71399689086081521408986944291200412162688643892096
8531441477697510777617942013838289517858628278242128130208059
91968300242461007242600083084500447195001248870018900001542006300100240983700
10714492911973276746751508169082893047930641017500121135693427415272027043590751211979655106
112550916857847060829243998085489242704481504320022365414725905690569189892884823204514401267214416760621520
1289813529274553597517839440073420855254824599915362926649627576886271557617716344362173579981631272572315117238169146277857557
1331248730812840850036106631812560313545019060422246164340300606213908123415131840308254186244827598776479110632298217921118572229321961315760770576
14107616802559321809575626785614225228203027325037529864190003245390673525162016346992549553875800096347072250121418797509950440505221125016789502992522512192831101925
Technical details
Node id
n-c823eaf24c675454 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-322b0642acfc2857

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:      12,        4;
  2:      81,       63,        9;
  3:     432,      592,      192,       16;
  4:    2025,     4375,     2425,      450,       25;
  5:    8748,    28116,    23760,     7380,      900,       36;
  6:   35721,   164983,   199969,    92610,    18620,     1617,      49;
  7:  139968,   908608,  1521408,   986944,   291200,    41216,    2688,     64;
  8:  531441,  4776975, 10777617,  9420138,  3828951,   785862,   82782,   4212,   81;
  9: 1968300, 24246100, 72426000, 83084500, 44719500, 12488700, 1890000, 154200, 6300, 100;
%F Formula
The row polynomials satisfy n^2*P(n,x) = (3*n^2+6*n+3+(n^2+2*n+1)*x)*P(n-1,x) + (n^2+2*n+1)*x*d/dx 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[((3*n^2 + 6*n + 3 + (n^2 + 2*n + 1)*x)*P[n - 1, x] + (n^2 + 2*n + 1)*x*D[P[n - 1, x], x])/(n^2)]];
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] = ((3*n^2 + 6*n + 3 + (n^2 + 2*n + 1)*x)*v[i - 1] + (n^2 + 2*n + 1)*x*deriv(v[i - 1], x))/(n^2)); 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

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

Proof chains ? JSON

Same fingerprint ?

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  8. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.