Real-rooted atlas

A131431 → x ↦ x+1 → basis A008277

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
18412
22835770
3801909010370
4208845715182131963
55123376456020003201610784
6121612635256691729046555131961997
728164529813351812982252360954877022374154
86400157625658375889350504525135450178501100252370700
91433653678831260605717740438648016269963116403108015122815752660
1031744179877514438281351127703546911517465679456155765472051150201531109585837
11696325953366653149862083894342719204401729929525917989011465718127908080410261834796648560
12151552195117652907909551204961758200225725716137126727034701961766982362639679923544951216933330376040430745
13327680634452401278568200682767540014290276000143764821207828620800247973440047639592056536480412412017836041604047682368920
14704512204963155556604689338073216090995120312551235450746538278085715932407197680778697062011806694901139778646926010254345511743391178894886
Technical details
Node id
n-1b94fba2375d1bc1 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-b5f1389370a379f1

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:     8,      4;
  2:    28,     35,       7;
  3:    80,    190,      90,      10;
  4:   208,    845,     715,     182,      13;
  5:   512,   3376,    4560,    2000,     320,      16;
  6:  1216,  12635,   25669,   17290,    4655,     513,     19;
  7:  2816,  45298,  133518,  129822,   52360,    9548,    770,    22;
  8:  6400, 157625,  658375,  889350,  504525,  135450,  17850,  1100,   25;
  9: 14336, 536788, 3126060, 5717740, 4386480, 1626996, 311640, 31080, 1512, 28;
%F Formula
The row polynomials satisfy (3*n-2)*P(n,x) = (6*n+2+(3*n+1)*x)*P(n-1,x) + (3*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[((6*n + 2 + (3*n + 1)*x)*P[n - 1, x] + (3*n + 1)*x*D[P[n - 1, x], x])/(3*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] = ((6*n + 2 + (3*n + 1)*x)*v[i - 1] + (3*n + 1)*x*deriv(v[i - 1], x))/(3*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.