Real-rooted atlas

A038291 → basis A013610 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(10 + 3\,n x\right)\,P_{n}(x) = \left(100 + 60\,x + 9\,x^{2} + 30\,n x + 9\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
110616
210012027247
3100018008101083718
4100002400016200432040554925
5100000300000270000108000202501458799708
6100000036000004050000216000060750087480510311510083
71000000042000000567000003780000014175000306180035721017496164111506
810000000048000000075600000060480000028350000081648000142884001399680590492321695129
910000000005400000000972000000090720000005103000000183708000042865200062985600531441019683032629228840
101000000000060000000000121500000000129600000000850500000003674160000010716300000209952000026572050019683000649539455993473039
1110000000000066000000000014850000000001782000000000133650000000067359600000023575860000057736800000974308500010825650007144929021257646341490625054
12100000000000072000000000001782000000000023760000000000200475000000001154736000000047151720000001385683200000292292550000433026000004286957400255091680690873387815859307813
13100000000000007800000000000021060000000000030888000000000028957500000000018764460000000087567480000000300231360000007599606300000140733450000018576815400016580959200898135290223205221211500426369012
14100000000000000840000000000000245700000000000039312000000000004054050000000000291891600000000015324309000000006004627200000001773241470000003940536600000065018853900007737780960006286947030031248730807174453516658130862573915
Technical details
Node id
n-cea1c0a6f6924fc7 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-710d46ba25b489cc

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:         10,          6;
  2:        100,        120,         27;
  3:       1000,       1800,        810,        108;
  4:      10000,      24000,      16200,       4320,        405;
  5:     100000,     300000,     270000,     108000,      20250,       1458;
  6:    1000000,    3600000,    4050000,    2160000,     607500,      87480,      5103;
  7:   10000000,   42000000,   56700000,   37800000,   14175000,    3061800,    357210,    17496;
  8:  100000000,  480000000,  756000000,  604800000,  283500000,   81648000,  14288400,  1399680,   59049;
  9: 1000000000, 5400000000, 9720000000, 9072000000, 5103000000, 1837080000, 428652000, 62985600, 5314410, 196830;
%F Formula
The row polynomials satisfy (10+3*n*x)*P(n,x) = (100+(30*n+60)*x+(9*n+9)*x^2)*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[Together[((100 + (30*n + 60)*x + (9*n + 9)*x^2)*P[n - 1, x])/(10 + 3*n*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] = ((100 + (30*n + 60)*x + (9*n + 9)*x^2)*v[i - 1])/(10 + 3*n*x)); 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.