Real-rooted atlas

A038335 → x ↦ x+1 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1301848
290010803242304
32700048600291605832110592
4810000194400017496006998401049765308416
524300000729000008748000052488000157464001889568254803968
672900000026244000003936600000314928000014171760003401222403401222412230590464
7218700000009185400000016533720000016533720000099202320000357128352007142567040612220032587068342272
8656100000000314928000000066134880000007936185600000595213920000028570268160008571080448001469328076801101996057628179280429056
919683000000000106288200000000255091680000000357128352000000321415516800000192849310080000771397240320001983592903680029753893555201983592903681352605460594688
10590490000000000354294000000000095659380000000001530550080000000016070775840000000115709586048000005785479302400000198359290368000044630840332800059507787110400357046722662464925062108545024
11177147000000000001169170200000000003507510600000000006313519080000000007576222896000000006364027232640000003818416339584000001636464145536000004909392436608000098187848732160001178254184785920642684100792323116402981210161152
12531441000000000000382637520000000000012627038160000000000252540763200000000003409300303200000000032729282910720000000229104980375040000001178254184785920000044184531929472000001178254184785920000212085753261465600231366276285235201156831381426176149587343098087735296
1315943230000000000000124357194000000000000447685898400000000000984908976480000000000147736346472000000000015955525418976000000001276442033518080000000765865220110848000000344639349049881600000114879783016627200000275711479239905280004511642387562086400451164238756208640208229648656711687180192468708211294208
1447829690000000000000040176939600000000000001566900644400000000000037605615465600000000000620492655182400000000007445911862188800000000067013206759699200000000459519132066508800000002412475443349171200000096499017733966848000002894970532019005440000631629934258692096000947444901388038144008745645243581890560374813367582081024344649238497994142121984
Technical details
Node id
n-70416a49496e0b07 · revision 0 · JSON
Definition source
derived:basis_map_A013609
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:             30,              18;
  2:            900,            1080,             324;
  3:          27000,           48600,           29160,            5832;
  4:         810000,         1944000,         1749600,          699840,          104976;
  5:       24300000,        72900000,        87480000,        52488000,        15746400,         1889568;
  6:      729000000,      2624400000,      3936600000,      3149280000,      1417176000,       340122240,       34012224;
  7:    21870000000,     91854000000,    165337200000,    165337200000,     99202320000,     35712835200,     7142567040,      612220032;
  8:   656100000000,   3149280000000,   6613488000000,   7936185600000,   5952139200000,   2857026816000,   857108044800,   146932807680,   11019960576;
  9: 19683000000000, 106288200000000, 255091680000000, 357128352000000, 321415516800000, 192849310080000, 77139724032000, 19835929036800, 2975389355520, 198359290368;
%F Formula
The row polynomials satisfy P(n,x) = (30+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[(30 + 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] = (30 + 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

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.