Real-rooted atlas

A038255 → basis A013610 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(7 + 3\,n x\right)\,P_{n}(x) = \left(49 + 42\,x + 9\,x^{2} + 21\,n x + 9\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
17613
2498427160
33438825671081900
4240182327938302440522000
516807720309261052920141751458250000
61176496050529724057408802976756123651032800000
7823543494125895295699075780486202515002822500471749631000000
8576480139530064889426441016487366806835028005264700131697977659049340000000
94035360731129925480048379610673117288576612104410829081470276363086294437200871968303700000000
10282475249242121642070042332151067311728010006047450617516071225729836307201353601302030451377810064953940000000000
11197732674318643366434599251063951027287538201100665219507924789580439623947902138626056803341878155530456850500145032125764430000000000
121384128720114236752549650337089371895880170232011556984804759509747496485547352706282328917754247017944125514852791800210060912617856417669087334600000000000
139688901040710796204016784164250120758872509549111211685395747025108173377772467211558518164353219192726412772658308413379010134506371847682281246700086286947032232052249000000000000
14678223072849813867687418834008042652857777326689208161145168783208451177887891300128834159184750949450686981696208620085704036622859863620156110268213926540588692830806040447218741115671744535520000000000000
Technical details
Node id
n-8d0909a4318144f1 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-5520686a41bcab10

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:        7,         6;
  2:       49,        84,        27;
  3:      343,       882,       567,        108;
  4:     2401,      8232,      7938,       3024,       405;
  5:    16807,     72030,     92610,      52920,     14175,      1458;
  6:   117649,    605052,    972405,     740880,    297675,     61236,      5103;
  7:   823543,   4941258,   9529569,    9075780,   4862025,   1500282,    250047,    17496;
  8:  5764801,  39530064,  88942644,  101648736,  68068350,  28005264,   7001316,   979776,   59049;
  9: 40353607, 311299254, 800483796, 1067311728, 857661210, 441082908, 147027636, 30862944, 3720087, 196830;
%F Formula
The row polynomials satisfy (7+3*n*x)*P(n,x) = (49+(21*n+42)*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[((49 + (21*n + 42)*x + (9*n + 9)*x^2)*P[n - 1, x])/(7 + 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] = ((49 + (21*n + 42)*x + (9*n + 9)*x^2)*v[i - 1])/(7 + 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.