Real-rooted atlas

A038326 → x ↦ x+1 → basis A007318

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(35 + 12\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1351247
212258401442209
34287544100151201728103823
4150062520580001058400241920207364879681
5525218759003750061740000211680003628800248832229345007
6183826562537815750003241350000148176000038102400052254720298598410779215329
7643392968751544143125001588261500009075780000031116960000640120320073156608035831808506623120463
8225187539062561765725000007411887000000508243680000021781872000005974456320001024192512001003290624042998169623811286661761
9788156386718752432025421875003335349150000002668279320000001372257936000004704884352000010754021376000158018273280013544423424051597803521119130473102767
10275854735351562594578766406250001459215253125000013341396600000000800483796000000032934190464000009409768704000001843546521600002370274099200018059231232006191736422452599132235830049
119654915737304687536412825066406250062421985828125000064205471137500000044026608780000000021132772214400000072455219020800000177441352704000003041851760640000347640201216000238381852262407430083706882472159215084012303
1233792205080566406251390307866171875000026217234047812500000299625531975000000002311396960950000000012679663328640000000507186533145600000014905073627136000003193944348672000004866962817024000050060188975104003120635156889608916100448256116191483108948578241
131182727177819824218755271583992568359375001084440135614062500000136329617048625000000011685395747025000000007211558518164000000003296712465446400000001130301416724480000002906489357291520000055361702043648000007592461994557440007099444981923840040568257039564801069932053790725460999706120583177327
144139545122369384765625198698165873730468750004428130553757421875000060728647594387500000000572584391604225000000003926292971000400000000020192363850859200000000791210991707136000000023736329751214080000005425446800277504000009300765943332864000011595760137142272000993922297469337600524266706357452801283918464548864256666986187667409334369
Technical details
Node id
n-fb74ce37c1c8f6e0 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-5ffbd82c13020df8

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:             35,              12;
  2:           1225,             840,             144;
  3:          42875,           44100,           15120,            1728;
  4:        1500625,         2058000,         1058400,          241920,           20736;
  5:       52521875,        90037500,        61740000,        21168000,         3628800,         248832;
  6:     1838265625,      3781575000,      3241350000,      1481760000,       381024000,       52254720,        2985984;
  7:    64339296875,    154414312500,    158826150000,     90757800000,     31116960000,     6401203200,      731566080,      35831808;
  8:  2251875390625,   6176572500000,   7411887000000,   5082436800000,   2178187200000,   597445632000,   102419251200,   10032906240,    429981696;
  9: 78815638671875, 243202542187500, 333534915000000, 266827932000000, 137225793600000, 47048843520000, 10754021376000, 1580182732800, 135444234240, 5159780352;
%F Formula
The row polynomials satisfy P(n,x) = (35+12*x)*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[(35 + 12*x)*P[n - 1, 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] = (35 + 12*x)*v[i - 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

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.