Real-rooted atlas

A038221 → basis A013610 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(15 + 18\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1151833
22255403241089
333751215014580583235937
4506252430004374003499201049761185921
5759375455625010935000131220007873200188956839135393
61139062582012500246037500393660000354294000170061120340122241291467969
71708593751435218750516678750010333575000124002900008928208800357128352061222003242618442977
825628906252460375000010333575000024800580000037200870000035712835200021427701120073466403840110199605761406408618241
93844335937541518828125019929037500005580130500000100442349000001205308188000096424655040004958982259200148769467776019835929036846411484401953
105766503906256919804687500373669453125001195742250000002511058725000003615924564000003615924564000002479491129600001115771008320002975389355520035704672266241531578985264449
1186497558593751141767773437506850606640625002466218390625000591892413750000099437925510000001193255106120000010227900909600000613674054576000024546962183040005891270923929606426841007923250542106513726817
1212974633789062518683472656250001233109195312500049324367812500000133175793093750000255697522740000000357976531836000000368204432745600000276153324559200000147281773098240000530214383153664001156831381426176011568313814261761667889514952984961
1319461950683593753036064306640625021859663007812500096182517234375000028854755170312500006232627116787500000997220338686000000011966644064232000000107699796578088000007179986438539200000344639349049881600011279105968905216002255821193781043202082296486567116855040353993448503713
142919292602539062549044115722656250038254410263671875001836211692656250000060594985857656250000145427966058375000000261770338905075000000358999321926960000000376949288023308000000301559430418646400000180935658251187840000789537417823365120002368612253470095360043728226217909452803748133675820810241816331681783800622529
Technical details
Node id
n-4739a5ec94dc3fc8 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-da2debf48cb6165b

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:          15,           18;
  2:         225,          540,           324;
  3:        3375,        12150,         14580,          5832;
  4:       50625,       243000,        437400,        349920,         104976;
  5:      759375,      4556250,      10935000,      13122000,        7873200,        1889568;
  6:    11390625,     82012500,     246037500,     393660000,      354294000,      170061120,      34012224;
  7:   170859375,   1435218750,    5166787500,   10333575000,    12400290000,     8928208800,    3571283520,     612220032;
  8:  2562890625,  24603750000,  103335750000,  248005800000,   372008700000,   357128352000,  214277011200,   73466403840,   11019960576;
  9: 38443359375, 415188281250, 1992903750000, 5580130500000, 10044234900000, 12053081880000, 9642465504000, 4958982259200, 1487694677760, 198359290368;
%F Formula
The row polynomials satisfy P(n,x) = (15+18*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[(15 + 18*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] = (15 + 18*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

Candidate equalities ?

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.