Real-rooted atlas

A038279 → basis A013611 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(13 + 12\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1131225
2169312144625
3219760845616172815625
4285611054561460168985620736390625
537129317136603163680292032013478402488329765625
6482680926733096616917607592832052565760194088962985984244140625
762748517405451956112279003217273692801594494720883104768271724544358318086103515625
8815730721602385763219461693888359292810244145686272030614298624141296762883726508032429981696152587890625
910604499373880989178683252883121287006209799689700905876488954682347525510573752322180007198725030785843251597803523814697265625
101378584918491272539924760528593507208013011532485120210186293990402328217410355217909364695040944669786112032700107980806707714457606191736422495367431640625
11179216039403718197320924068839876350341602325811431715204293805720089605548918161346565122078302781443377194485350401558705147084804796015837184088541830840327430083706882384185791015625
122329808512248125807109674132813102071065328964031406481639680837292115417472012366160473858048133174035872317441053684679429324860789500736307202493928235335680690626280554496115909305827328891610044825659604644775390625
133028751065922533634501279107036201295455458235846813076953971059215722485278394752026123514001025126432152017232030924829678785137259315220546851248871833610536846794293248038905280471236608979433634240921615068209757552641069932053790721490116119384765625
143937376385699289508830179074985043052981074449910241127254550566120704286149232066784486452827550535406366727314583920287035392771648413568742195262325448788244561923835412233122742272177019026144126566459418973810615910413712070879372902419472763378991104128391846454886437252902984619140625
Technical details
Node id
n-fbee24f9afac6b9c · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-42b2d489dcb16cdd

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:          13,          12;
  2:         169,         312,          144;
  3:        2197,        6084,         5616,         1728;
  4:       28561,      105456,       146016,        89856,        20736;
  5:      371293,     1713660,      3163680,      2920320,      1347840,       248832;
  6:     4826809,    26733096,     61691760,     75928320,     52565760,     19408896,      2985984;
  7:    62748517,   405451956,   1122790032,   1727369280,   1594494720,    883104768,    271724544,     35831808;
  8:   815730721,  6023857632,  19461693888,  35929281024,  41456862720,  30614298624,  14129676288,   3726508032,   429981696;
  9: 10604499373, 88098917868, 325288312128, 700620979968, 970090587648, 895468234752, 551057375232, 218000719872, 50307858432, 5159780352;
%F Formula
The row polynomials satisfy P(n,x) = (13+12*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[(13 + 12*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] = (13 + 12*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.