Real-rooted atlas

A038314 → x ↦ x+1 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(44 + 24\,x + 22\,n + 12\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 528 + 288\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8Pn(1)
0528288816
11742419008518441616
2511104836352456192829441886592
31405536030666240250905609123840124416080180160
437106150410119859201103984640602173440164229120179159043271350528
59523911936311691663364250340864030911569920126456422402759049216250822656129763570944
6239458357248914295545856149611998412813601090764807418776780802427963310084414478745634398535685042241613824
759265943418882586150258278449371959476224538603194286083672294506496016024557846528437033395814468109100646446438023168192865741728768

All 15 rows

Technical details
Node id
n-87628f1b2cb96ac0 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-57ca34e86b523c55

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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:              528,               288;
  1:            17424,             19008,              5184;
  2:           511104,            836352,            456192,             82944;
  3:         14055360,          30666240,          25090560,           9123840,           1244160;
  4:        371061504,        1011985920,        1103984640,         602173440,         164229120,          17915904;
  5:       9523911936,       31169166336,       42503408640,       30911569920,       12645642240,        2759049216,         250822656;
  6:     239458357248,      914295545856,     1496119984128,     1360109076480,      741877678080,      242796331008,       44144787456,       3439853568;
  7:    5926594341888,    25861502582784,    49371959476224,    53860319428608,    36722945064960,    16024557846528,     4370333958144,     681091006464,      46438023168;
  8:  144872306135040,   711191321026560,  1551690154967040,  1974878379048960,  1615809582858240,   881350681559040,   320491156930560,   74920010711040,   10216365096960,    619173642240;
  9: 3505909808467968, 19123144409825280, 46938627187752960, 68274366818549760, 65170986508615680, 42657372987457536, 19389714994298880, 6043547530690560, 1236180176732160, 149840021422080, 8173092077568;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (22*n+44+(12*n+24)*x)*P(n-1,x) for n >= 1, with P(0,x) = 528 + 288*x.
%t Mathematica
P[0, x_] := 528 + 288*x;
P[n_, x_] := P[n, x] = Expand[Together[((22*n + 44 + (12*n + 24)*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] = 528 + 288*x; for(n = 1, N, my(i = n + 1); v[i] = ((22*n + 44 + (12*n + 24)*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. [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. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.