Real-rooted atlas

A139548 → basis A013610 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(2 - 3\,x + 6\,n x\right)\,P_{n}(x) = \left(16 + 72\,x + 108\,x^{2} + 54\,x^{3} + 48\,n x + 144\,n x^{2} + 108\,n x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
184854110
2647682592345616208500
351292165184013824019440013996840824575000
440969830477414430965767257600104509449144576447897694478436250000
532768983040995328053084160174182400376233984548674560537477120340122240125971200207852482187500000
626214494371841167851527785676803284582400945959731219313344512283787919362993075712022170931200109746109443265173504442158912128125000000
7209715288080384128817561610305404928531372441601912940789765021469573129837981204481452639412224161404379136013315861278727923487703043218916879367999675084891833004807343750000000

All 15 rows

Technical details
Node id
n-aef5b6c97c5c498f · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-875f7e3274313d29

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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:         8,         48,           54;
  2:        64,        768,         2592,          3456,           1620;
  3:       512,       9216,        51840,        138240,         194400,         139968,           40824;
  4:      4096,      98304,       774144,       3096576,        7257600,       10450944,         9144576,         4478976,           944784;
  5:     32768,     983040,      9953280,      53084160,      174182400,      376233984,       548674560,       537477120,        340122240,        125971200,         20785248;
  6:    262144,    9437184,    116785152,     778567680,     3284582400,     9459597312,     19313344512,     28378791936,      29930757120,      22170931200,      10974610944,       3265173504,        442158912;
  7:   2097152,   88080384,   1288175616,   10305404928,    53137244160,   191294078976,    502146957312,    983798120448,    1452639412224,    1614043791360,    1331586127872,     792348770304,     321891687936,      79996750848,       9183300480;
  8:  16777216,  805306368,  13589544960,  126835752960,   772905369600,  3338951196672,  10712468422656,  26234616545280,   49804779847680,   73784859033600,   85221512183808,   76065481949184,   51502670069760,   25598960271360,    8815968460800,   1880739938304,    187339329792;
  9: 134217728, 7247757312, 138613358592, 1478542491648, 10396001894400, 52395849547776, 198667596201984, 583839466389504, 1354690011856896, 2508685207142400, 3725397532606464, 4433530947895296, 4202617877692416, 3133312737214464, 1798457566003200, 767341894828032, 229303339665408, 42845606719488, 3768826516992;
%F Formula
The row polynomials satisfy (2+(6*n-3)*x)*P(n,x) = (16+(48*n+72)*x+(144*n+108)*x^2+(108*n+54)*x^3)*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[((16 + (48*n + 72)*x + (144*n + 108)*x^2 + (108*n + 54)*x^3)*P[n - 1, x])/(2 + (6*n - 3)*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 + (48*n + 72)*x + (144*n + 108)*x^2 + (108*n + 54)*x^3)*v[i - 1])/(2 + (6*n - 3)*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. [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
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.