Real-rooted atlas

A137597 → reverse → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(1 - x - x^{2} + 2\,n + 3\,n x + n x^{2}\right)\,P_{n-1}(x) + \left(-2 - 5\,x - 4\,x^{2} - x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1325
21115531
347986715227
4227641670307521897
51215434461464299148620317693
67107308235519152204275017648877181351
7449592296064985415963994244111796124184741402019515
83050911795165458862066532615983741341741312100982428572114724217393
9220639914710196433123727390443180519503580780612772836684483711490395115975310514069
101691398712609921942062199782651194210593008249251274895574675762288305606122841496421576785704232422735
111368232631128545538420893231193675098431382184239814194530888103514339405359756709193075068046079282665566810421359761030698803
12116349049910523995721434188931781080269870231804970202982133378770261828770964801145433806985201993559516704086156359958983046734369227644437927207340633
131036625203110206497211246173700239012706784301632372990966204317532792605531318193296402304978474798126576367231751217116965214841578737829163988944348285108719089932214789151722093
1496491364675102762765439950600977089791526767103686431532420936303471510382052705263728716617944561462406460287449740306311405868943764951313349181391355248327354244823629233270756446871382958545246897625425367
Technical details
Node id
n-2f22ddcafff9670e · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-e49234ad2c21a445

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:       3,        2;
  2:      11,       15,        5;
  3:      47,       98,       67,       15;
  4:     227,      641,      670,      307,       52;
  5:    1215,     4344,     6146,     4299,     1486,      203;
  6:    7107,    30823,    55191,    52204,    27501,     7648,      877;
  7:   44959,   229606,   498541,   596399,   424411,   179612,    41847,    4140;
  8:  305091,  1795165,  4588620,  6653261,  5983741,  3417413,  1210098,  242857,   21147;
  9: 2206399, 14710196, 43312372, 73904431, 80519503, 58078061, 27728366, 8448371, 1490395, 115975;
%F Formula
The row polynomials satisfy P(n,x) = (2*n+1+(3*n-1)*x+(n-1)*x^2)*P(n-1,x) - (2+5*x+4*x^2+x^3)*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[(2*n + 1 + (3*n - 1)*x + (n - 1)*x^2)*P[n - 1, x] - (2 + 5*x + 4*x^2 + x^3)*D[P[n - 1, x], x]];
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] = (2*n + 1 + (3*n - 1)*x + (n - 1)*x^2)*v[i - 1] - (2 + 5*x + 4*x^2 + x^3)*deriv(v[i - 1], x)); 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

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.