Real-rooted atlas

A038259 → basis A013611 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(20 + 11\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1201131
2400440121961
38000132007260133129791
416000035200029040010648014641923521
53200000880000096800005324000146410016105128629151
66400000021120000029040000021296000087846000193261201771561887503681
71280000000492800000081312000007453600000409948000013528284002480185401948717127512614111
82560000000011264000000021683200000023851520000016397920000072150848000198414832003117947360214358881852891037441
951200000000025344000000005575680000000715545600000059032512000003246788160000119048899200028061526240038584598580235794769126439622160671
1010240000000000563200000000001393920000000002044416000000001967750400000001298715264000005952444960000018707684160000385845985800047158953820025937424601819628286980801
11204800000000000123904000000000034073600000000005622144000000000618435840000000047619559680000002619075782400000102892262880000028295372292000051874849202000570623341222028531167061125408476896404831
124096000000000000270336000000000008177664000000000014992384000000000018553075200000000016326706176000000010476303129600000049388286182400000169772233752000004149987936160000684748009466400684748009466403138428376721787662783788549761
13819200000000000005857280000000000001932902400000000000389801984000000000053597772800000000005306179507200000000389119830528000000021401590679040000008828156155104000002697492158504000005934482748708800089017241230632008159913779474603452271214393124417546297445042591
14163840000000000000012615680000000000000451010560000000000009922232320000000000015007376384000000000016508114022400000000013619194068480000000085606362716160000000411980620571520000001510595608762240000041541379240961600008308275848192320001142387929126444009666359400300680379749833583241756943935220796320321
Technical details
Node id
n-d78ca5ecb0198bfb · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-590b4d5bb5323b01

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:            1;
  1:           20,            11;
  2:          400,           440,           121;
  3:         8000,         13200,          7260,          1331;
  4:       160000,        352000,        290400,        106480,         14641;
  5:      3200000,       8800000,       9680000,       5324000,       1464100,        161051;
  6:     64000000,     211200000,     290400000,     212960000,      87846000,      19326120,       1771561;
  7:   1280000000,    4928000000,    8131200000,    7453600000,    4099480000,    1352828400,     248018540,     19487171;
  8:  25600000000,  112640000000,  216832000000,  238515200000,  163979200000,   72150848000,   19841483200,   3117947360,   214358881;
  9: 512000000000, 2534400000000, 5575680000000, 7155456000000, 5903251200000, 3246788160000, 1190488992000, 280615262400, 38584598580, 2357947691;
%F Formula
The row polynomials satisfy P(n,x) = (20+11*x)*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 + 11*x)*P[n - 1, 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 + 11*x)*v[i - 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.