Real-rooted atlas

A013622 → basis A013609 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(18 + 30\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1183048
232410809002304
35832291604860027000110592
4104976699840174960019440008100005308416
518895681574640052488000874800007290000024300000254803968
634012224340122240141717600031492800003936600000262440000072900000012230590464
7612220032714256704035712835200992023200001653372000001653372000009185400000021870000000587068342272
8110199605761469328076808571080448002857026816000595213920000079361856000006613488000000314928000000065610000000028179280429056
919835929036829753893555201983592903680077139724032000192849310080000321415516800000357128352000000255091680000000106288200000000196830000000001352605460594688
10357046722662459507787110400446308403328000198359290368000057854793024000001157095860480000016070775840000000153055008000000009565938000000000354294000000000059049000000000064925062108545024
11642684100792321178254184785920981878487321600049093924366080000163646414553600000381841633958400000636402723264000000757622289600000000631351908000000000350751060000000000116917020000000000177147000000000003116402981210161152
12115683138142617623136627628523520212085753261465600117825418478592000044184531929472000001178254184785920000022910498037504000000327292829107200000003409300303200000000025254076320000000000126270381600000000003826375200000000000531441000000000000149587343098087735296
1320822964865671168451164238756208640451164238756208640027571147923990528000114879783016627200000344639349049881600000765865220110848000000127644203351808000000015955525418976000000001477363464720000000000984908976480000000000447685898400000000000124357194000000000000159432300000000000007180192468708211294208
1437481336758208102487456452435818905609474449013880381440063162993425869209600028949705320190054400009649901773396684800000241247544334917120000004595191320665088000000067013206759699200000000744591186218880000000006204926551824000000000037605615465600000000000156690064440000000000004017693960000000000000478296900000000000000344649238497994142121984
Technical details
Node id
n-f9c8690cbff4f27f · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-b1029f17d1b64336

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:           18,            30;
  2:          324,          1080,            900;
  3:         5832,         29160,          48600,          27000;
  4:       104976,        699840,        1749600,        1944000,          810000;
  5:      1889568,      15746400,       52488000,       87480000,        72900000,        24300000;
  6:     34012224,     340122240,     1417176000,     3149280000,      3936600000,      2624400000,       729000000;
  7:    612220032,    7142567040,    35712835200,    99202320000,    165337200000,    165337200000,     91854000000,     21870000000;
  8:  11019960576,  146932807680,   857108044800,  2857026816000,   5952139200000,   7936185600000,   6613488000000,   3149280000000,    656100000000;
  9: 198359290368, 2975389355520, 19835929036800, 77139724032000, 192849310080000, 321415516800000, 357128352000000, 255091680000000, 106288200000000, 19683000000000;
%F Formula
The row polynomials satisfy P(n,x) = (18+30*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[(18 + 30*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] = (18 + 30*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.