Real-rooted atlas

A013609 → basis A013610 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(3\,n + 6\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 6\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
166
23672108
31626486481458
464838887776518417496
5243019440583207776038880196830
68748874803499206998406998402799362125764
7306183674161837080489888073483205878656195955222320522
8104976146966488179842939328058786560705438724702924813436928229582512
9354294566870439680928158723712396809280634894848634894848362797056906992642324522934
101180980212576401700611207936185602380855680476171136063489484805441955840272097792060466176023245229340
1138972347794468070150212037413446401309470624031427294976523788249605986151424044896135680199538380803990767616230127770466
121275458428060084828060084801683605088067344203520188563769856377127539712538753628160538753628160359169085440143667634176261213880322259436291848
13414523989948575521094343307272956220480328302992160105056957491224513290081284202278299648525284787456046691981107202801518866432101873413324816978902220822029503845518
141339231323482001432417840171843064161260161532080630080551549026828814707974048768294159480975364412392214630449026580162560392212641300482139341679820871311389327361097098297344213516729579636
Technical details
Node id
n-6e3adcccc1f149b8 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-11aec07ce60b6d61

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:      0;
  1:      6;
  2:     36,      72;
  3:    162,     648,      648;
  4:    648,    3888,     7776,      5184;
  5:   2430,   19440,    58320,     77760,     38880;
  6:   8748,   87480,   349920,    699840,    699840,    279936;
  7:  30618,  367416,  1837080,   4898880,   7348320,   5878656,   1959552;
  8: 104976, 1469664,  8817984,  29393280,  58786560,  70543872,  47029248,  13436928;
  9: 354294, 5668704, 39680928, 158723712, 396809280, 634894848, 634894848, 362797056, 90699264;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (3*n+6*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 6.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 6;
P[n_, x_] := P[n, x] = Expand[Together[((3*n + 6*n*x)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 6; for(n = 2, N, my(i = n + 1); v[i] = ((3*n + 6*n*x)*v[i - 1])/(n - 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.