Real-rooted atlas

A059364 → basis A013611 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-n - 8\,n x - 16\,n x^{2} + 4\,n^{2} + 24\,n^{2} x + 32\,n^{2} x^{2} - 4\,n^{3} - 16\,n^{3} x - 16\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-1 - 2\,n - 8\,n x + 4\,n^{2} + 8\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
13811
2158896199
3105976225615364873
494512144437286195230720149489
510395171384845328184755218867207372805498659
613513527269521715908851068672799539206392217620643840235609783
7202702548471840371918304140663603229867123203645284352239778201666060288011519434769
8344594259537508808649259488397813268481070395507201766595870721768529100809892056268823781703680632673110881
965472907520602345320215838477792117100966553638239111718407980640745472107799079485449155724705792445813515878495126814720038557693095355
1013749310575485027577720576874875283236102588458496138852617885440351141191903232598807968374784685066010755072505628450095104218115971481600418557984768002581838123071655
1131623414322512365573618160164693041444656116879279471155251792525765824001540766563812556831797831123652608458680046570700804554022188456345629736280459837440115194963846758402009078326886400188403998695311385
1279058535806253394978182036005007647336623920397601745202467841996088820972531206835165893642854401657775053826347008289103828092426649636153777985958707203172218845339320320185806301789631283265342967795154944010447207299809280014880615444521953105
1321345804667687599886498638822001616873079087266401420721594052588288797385116044187520030910542155704301568862265995414113525761765772367837781032962667409399961580994562945384038541452902402316186194366095687681230454488653553991683963061782888579072058504360878931968001264705306720254851795
14619028335362937531357814728039380055279785179191773605327357855137664947233069219523649556096014322485204971187589124520060140873610260480106392422373798915604481885985317726660539187225138827755975606272000248605056410770273730561771811004085909114060886140451482025821470722559693752537091932160351026165273591808000115083426100678642055575
Technical details
Node id
n-e3c916628355a668 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-ba66270f11403229

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:         1;
  1:         3,           8;
  2:        15,          88,           96;
  3:       105,         976,         2256,          1536;
  4:       945,       12144,        43728,         61952,         30720;
  5:     10395,      171384,       845328,       1847552,       1886720,        737280;
  6:    135135,     2726952,     17159088,      51068672,      79953920,      63922176,       20643840;
  7:   2027025,    48471840,    371918304,    1406636032,    2986712320,    3645284352,     2397782016,     660602880;
  8:  34459425,   953750880,   8649259488,   39781326848,  107039550720,  176659587072,   176852910080,   98920562688,   23781703680;
  9: 654729075, 20602345320, 215838477792, 1171009665536, 3823911171840, 7980640745472, 10779907948544, 9155724705792, 4458135158784, 951268147200;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(4*n^3-4*n^2+n+(16*n^3-24*n^2+8*n)*x+(16*n^3-32*n^2+16*n)*x^2)*P(n-2,x) + (4*n^2-2*n-1+(8*n^2-8*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 8*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 - 4*n^2 + n + (16*n^3 - 24*n^2 + 8*n)*x + (16*n^3 - 32*n^2 + 16*n)*x^2)*P[n - 2, x] + (4*n^2 - 2*n - 1 + (8*n^2 - 8*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] = 1; v[2] = 3 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^3 - 4*n^2 + n + (16*n^3 - 24*n^2 + 8*n)*x + (16*n^3 - 32*n^2 + 16*n)*x^2)*v[i - 2] + (4*n^2 - 2*n - 1 + (8*n^2 - 8*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

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.