Real-rooted atlas

A038334 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(5 + 2\,x + 2\,n\right)\,P_{n}(x) = \left(84 + 80\,x + 16\,x^{2} + 24\,n + 16\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 336 + 320\,x + 64\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
033632064720
151848064384051217600
276032165888129024409604096416000
31078272304128033177601720320409600327689600000
414929920517570567299072053084160206438403932160262144217600000
5203046912836075520144919756813624934407431782402312110083670016020971524864000000
62723217408129950023682675441664030916214784217998950409512681472246625075233554432016777216107520000000
7361184624641960716533764678200852486421059993605564918661123139184885761141521776642536715059230198988801342177282355200000000
847469979238428894769971207842866135040124752022732801284211998720089038698577924185579847680130459631616025367150592026843545600107374182451200000000000
96191736422400417735817297921271369878732802300574066278402744544500121602260213117747201305900912476165261871808512014350559477760248034361344023622320128085899345921105920000000000
10802449040343045944066965504002005131923030016406838361194496055213777590681605269525440233472361634098839552017909498228244486314246170214401530726344294402381129868902420615843020806871947673623756800000000000
111034267651997696834547001956761630909148220620800695112399983738881057779739105689601148446573886177289133844096404684853728494684733440232823476967178247296462241136640159195539806617622512500578713617867063951360549755813888507904000000000000
12132671574670049281158379770237419524673463210957864961153941533569843200194631471995446886423694266155967447042143766937920864256146141505542474956875219892558626816028973588244804403281720377100730368162090004166737922101166720679936153931627888640439804651110410813440000000000000
131694772373204500481592058896040591360695027862142451712018693852843831459840346182460070952960004671155327890725273647388532311934894080367502903643576729602192122583137124352010029319007816908800347683058937652838489149502291705856016209000416673792019395385113968640131941395333120035184372088832229376000000000000000
142156983020442091520216930863770176061441018917693465978470402965452211807793971205982032910026067148808862270977816395776009965131366166880583688665331622753809203205880046458297227673603117685451572799078401283752833000564326404045766504001778483295092802444486246401595963117949419520177329235327713280112589990684262402814749767106564849664000000000000000
Technical details
Node id
n-68e39df9a140191c · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-7ad822d7676abaf6

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:           336,            320,              64;
  1:          5184,           8064,            3840,             512;
  2:         76032,         165888,          129024,           40960,            4096;
  3:       1078272,        3041280,         3317760,         1720320,          409600,           32768;
  4:      14929920,       51757056,        72990720,        53084160,        20643840,         3932160,          262144;
  5:     203046912,      836075520,      1449197568,      1362493440,       743178240,       231211008,        36700160,        2097152;
  6:    2723217408,    12995002368,     26754416640,     30916214784,     21799895040,      9512681472,      2466250752,      335544320,       16777216;
  7:   36118462464,   196071653376,    467820085248,    642105999360,    556491866112,    313918488576,    114152177664,    25367150592,     3019898880,     134217728;
  8:  474699792384,  2889476997120,   7842866135040,  12475202273280,  12842119987200,   8903869857792,   4185579847680,  1304596316160,   253671505920,   26843545600,   1073741824;
  9: 6191736422400, 41773581729792, 127136987873280, 230057406627840, 274454450012160, 226021311774720, 130590091247616, 52618718085120, 14350559477760, 2480343613440, 236223201280, 8589934592;
%F Formula
The row polynomials satisfy (2*n+5+2*x)*P(n,x) = (24*n+84+(16*n+80)*x+16*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 336 + 320*x + 64*x^2.
%t Mathematica
P[0, x_] := 336 + 320*x + 64*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((24*n + 84 + (16*n + 80)*x + 16*x^2)*P[n - 1, x])/(2*n + 5 + 2*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 336 + 320*x + 64*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((24*n + 84 + (16*n + 80)*x + 16*x^2)*v[i - 1])/(2*n + 5 + 2*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. [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.