Real-rooted atlas

A038335 → basis A013609 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(21\,n + 18\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 18\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
11818
27566481404
323814408241749682134
4666792171460814696644199044270968
5175032906001128077157360440899209447840208209690
644108290818903553203240609120277766496011904278402040733449744213492
7108065312465557644640811909238516013610558304087496446240299987815684285540224443361713886
8259356749904155614049942440015041413765716434487680489980098944025198976517127199707576328815968460819761264961776
961273032164824201579348444812604738045334421608122363430423151559675104015875355205785668037236596224166621803909121785233613312867025500197922
1014297040838458011029145789667603781421413600320756284282720064097236550635436808334561483037440476260656173568017495289410457603748990587955203570467226624037571105008576620
113302616433683798283081408601468401091885433177092402495738132976211203743607199464316803850567405163297282750405289402355201347137284605235204330084129088256082477792935014407069525108715521611800404867936998
1275659940117119736713365149675700368305727921289585872078615751188750652801347698591807154048016172383101685848576138620426585878702088486964893012981760363727066843413504010392201909811814401781520327396311041388197657711411268574780861654046824
1317212636376644739941770442598740601822483463722512056943056238467778605876980160459902144454191318880630722940965748094464630722940965748094464463388283158508804096248243723120629716480945690373792875110402431775246895964569637897796055521525762706985432537251842897284491404883478314
143892703919025810417243375843669144744648822307576744131582962247010952691012783216704150234700521702497500802317906808049124247155226490363520561419967488227060258747669314007041459673091949302732902469508242473776320614402383139741958045278208557097602016166428672795853717165952040965247387146149134336121685948639005106089188
Technical details
Node id
n-86b3443645d7d7be · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-8e9037e0ccb64a62

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:             0;
  1:            18;
  2:           756,            648;
  3:         23814,          40824,           17496;
  4:        666792,        1714608,         1469664,          419904;
  5:      17503290,       60011280,        77157360,        44089920,         9447840;
  6:     441082908,     1890355320,      3240609120,      2777664960,      1190427840,       204073344;
  7:   10806531246,    55576446408,    119092385160,    136105583040,     87496446240,     29998781568,     4285540224;
  8:  259356749904,  1556140499424,   4001504141376,   5716434487680,   4899800989440,   2519897651712,   719970757632,    88159684608;
  9: 6127303216482, 42015793484448, 126047380453344, 216081223634304, 231515596751040, 158753552057856, 68037236596224, 16662180390912, 1785233613312;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (21*n+18*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 18.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 18;
P[n_, x_] := P[n, x] = Expand[Together[((21*n + 18*n*x)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 18; for(n = 2, N, my(i = n + 1); v[i] = ((21*n + 18*n*x)*v[i - 1])/(n - 1)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.