Real-rooted atlas

A038259 → basis A007318 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(16 + 10\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1161026
2256320100676
3409676804800100017576
4655361638401536006400010000456976
5104857632768004096000256000080000010000011881376
6167772166291456098304000819200003840000096000001000000308915776
72684354561174405120220200960022937600001433600000537600000112000000100000008031810176
84294967296214748364804697620480058720256000458752000002293760000071680000001280000000100000000208827064576
96871947673638654705664096636764160014092861440001321205760000825753600000344064000000921600000001440000000010000000005429503678976
101099511627776687194767360019327352832000322122547200003523215360000026424115200000137625600000004915200000000115200000000016000000000010000000000141167095653376
1117592186044416120946279055360377957122048000708669603840000885837004800000775107379200000484442112000000216268800000000675840000000001408000000000017600000000001000000000003670344486987776
1228147497671065621110623253299207256776743321600151182848819200002126008811520000021260088115200000155021475840000008304721920000000324403200000000090112000000000016896000000000019200000000000100000000000095428956661682176
1345035996273704963659174697238528013721905114644480031446032554393600049134425866240000055276229099520000046063524249600000028789702656000000013495173120000000046858240000000000117145600000000001996800000000000208000000000000100000000000002481152873203736576
14720575940379279366305039478318694402561422288066969600640355572016742400011006111394037760000137576392425472000001289778678988800000092127048499200000005038197964800000000209924915200000000065601536000000000014909440000000000023296000000000000224000000000000010000000000000064509974703297150976
Technical details
Node id
n-d742a8e69ac54bc3 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-4b28493176e02e0b

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:          16,           10;
  2:         256,          320,          100;
  3:        4096,         7680,         4800,          1000;
  4:       65536,       163840,       153600,         64000,         10000;
  5:     1048576,      3276800,      4096000,       2560000,        800000,       100000;
  6:    16777216,     62914560,     98304000,      81920000,      38400000,      9600000,      1000000;
  7:   268435456,   1174405120,   2202009600,    2293760000,    1433600000,    537600000,    112000000,    10000000;
  8:  4294967296,  21474836480,  46976204800,   58720256000,   45875200000,  22937600000,   7168000000,  1280000000,   100000000;
  9: 68719476736, 386547056640, 966367641600, 1409286144000, 1321205760000, 825753600000, 344064000000, 92160000000, 14400000000, 1000000000;
%F Formula
The row polynomials satisfy P(n,x) = (16+10*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[(16 + 10*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] = (16 + 10*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.