Real-rooted atlas

A147716 → basis A013610 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(18 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
118624
232421636576
358325832194421613824
410497613996869984155521296331776
518895683149280209952069984011664077767962624
634012224680244485668704025194240629856083980846656191102976
7612220032142851340814285134087936185602645395205290790458786562799364586471424
81101996057629386561536342843217922285621452895234227202539579392423263232403107841679616110075314176
9198359290368595077871104793437161472617117792256308558896128102852965376228562145283265173504272097792100776962641807540224
1035704672266241190155742208017852336133120158687432294409256766883840370270675353610285296537601959104102402448880128018139852806046617663403380965376
11642684100792322356508369571843927513949286403927513949286402618342632857601221893228666884072977428889696975653068801616260884480179584542720119723028483627970561521681143169024
1211568313814261764627325525704704848343013045862494260334782873607069525108715520377041339131494414662718744002564189348212572168727808776192012930087075840129300870758478364164096217678233636520347436056576
13208229648656711689023284775124172818046569550248345622056918339192422418380765282660352011028459169596211249015374087094272163384580290314244084614507257856756410093936640100854679191552916860719923250936706662413060694016876488338465357824
143748133675820810241749129048716378112378977960555215257650530394740695367684631952851230408704308796856748693913615439842837434695685881844890451312641715538093048299523812306873440665663538447890677767701630047354886418025039462432912948920327836416409621035720123168587776
Technical details
Node id
n-97c2a6af81d37e00 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-f9762973ec1089d7

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,            6;
  2:          324,          216,           36;
  3:         5832,         5832,         1944,          216;
  4:       104976,       139968,        69984,        15552,         1296;
  5:      1889568,      3149280,      2099520,       699840,       116640,         7776;
  6:     34012224,     68024448,     56687040,     25194240,      6298560,       839808,       46656;
  7:    612220032,   1428513408,   1428513408,    793618560,    264539520,     52907904,     5878656,     279936;
  8:  11019960576,  29386561536,  34284321792,  22856214528,   9523422720,   2539579392,   423263232,   40310784,   1679616;
  9: 198359290368, 595077871104, 793437161472, 617117792256, 308558896128, 102852965376, 22856214528, 3265173504, 272097792, 10077696;
%F Formula
The row polynomials satisfy P(n,x) = (18+6*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 + 6*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 + 6*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.