Real-rooted atlas

A027467 → basis A013611 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(20 + 4\,x\right)\,P_{n-1}(x) + \left(4\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
120424
240017616592
38000582411526415040
4160000171776555526656256394240
532000004762624224256043520035840102410677248
664000000127102976818421762289664030310401843204096299061248
7128000000033064714242799992832106006118420070400019496960917504163848667660288
825600000000844753141769162568499245121929216114655887361582694400118358016445644865536259994091520
951200000000021298075402242903420436480181040447488059324891136010917013094411538530304687341568212336642621448070298861568
1010240000000000531633809653768981500238233669546624942080285985787084806739800883200944375857152791465164803853516800996147201048576259130864435200
1120480000000000013168811431690242727473590566912258475200767590413077353332736003839863501619206851174124748875765362196485169689395202099249152046137344041943048602255129313280
12409600000000000032424347436056576816367851085496329362195860789657657417480028553216205903953395712004550462015537152637720703533056571885297336323243455283200111652372480211392921616777216295035695007793152
138192000000000000079456833846535782424155273727836160003322449815886888960244151711545950208010532857336970608642825819100419194884881244183173529655246863602810884103876147937281967296348160058190095974495965675526710886410446618066669273088
141638400000000000000193973201231685877767081303979180267929611598050359951491072010118441542008963072051897497809720442880166467469766327009283473324848090972160482533458061557760450804518695731202821928268070912115933515350016298017043251243218108416268435456381561803364391452672
Technical details
Node id
n-b1ad4f9d2fc1e443 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-73b18cc810b5ad72

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:           20,             4;
  2:          400,           176,            16;
  3:         8000,          5824,          1152,            64;
  4:       160000,        171776,         55552,          6656,          256;
  5:      3200000,       4762624,       2242560,        435200,        35840,         1024;
  6:     64000000,     127102976,      81842176,      22896640,      3031040,       184320,        4096;
  7:   1280000000,    3306471424,    2799992832,    1060061184,    200704000,     19496960,      917504,     16384;
  8:  25600000000,   84475314176,   91625684992,   45121929216,  11465588736,   1582694400,   118358016,   4456448,    65536;
  9: 512000000000, 2129807540224, 2903420436480, 1810404474880, 593248911360, 109170130944, 11538530304, 687341568, 21233664, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (20+4*x)*P(n-1,x) + 4*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[(20 + 4*x)*P[n - 1, x] + 4*x*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] = (20 + 4*x)*v[i - 1] + 4*x*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

  • 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.