Real-rooted atlas

A136124 → basis A013611 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(6 + 12\,x + n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
171219
256180144380
35042292345617287980
4504028968620645875220736175560
555440379128103032013910409331202488324037880
6665280521481616913376290563202788992014183424298598496909120
7864864075775968282451680580692672711244800519063552209018880358318082422728000
8121080960116464723248636351361151911756816925739264158018273289155026944300987187242998169662990928000
9181621440018922680000869302938242311503851523921154997764401362810883269473320961550084014084256818790451597803521700755056000
10290594304003245574528001617956861184474156968832090476526182401174756649472010512792686592640350240768025411918233605933747404806191736422447621141568000
114940103168005866189862400313999560737281000221670356482107089307699203082804618291203196882736087042350130531696641200422898892804058167246848081730920775687430083706881381013105472000
1288921857024001115195413248006355934876759042177198479526400499302675828633680775554821632009453754466906112806649424035840049809178560430082170977583104000634095727017984111451255603200891610044825641430393164160000
131689515283456002225577513600000134145107617397764899389296311244812099389016175718421338987526053683227655200065717452826670844416968294419143537014911795210101958835149209638099549810589696972672258067660815068209757552641069932053790721284342188088960000
143379030566912000465389686121472002949971453979955201140851988403126272300780451879249305657197241871518228488091718516269932544865279289127975321670292087330185543684317616208819257344197422605642969907265172904934060851214685709048322457620221715816644608128391846454886441098950018846720000
Technical details
Node id
n-233ab538ff185e07 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-6f88f43a262894ce

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:          7,          12;
  2:         56,         180,         144;
  3:        504,        2292,        3456,         1728;
  4:       5040,       28968,       62064,        58752,        20736;
  5:      55440,      379128,     1030320,      1391040,       933120,       248832;
  6:     665280,     5214816,    16913376,     29056320,     27889920,     14183424,      2985984;
  7:    8648640,    75775968,   282451680,    580692672,    711244800,    519063552,    209018880,     35831808;
  8:  121080960,  1164647232,  4863635136,  11519117568,  16925739264,  15801827328,   9155026944,   3009871872,   429981696;
  9: 1816214400, 18922680000, 86930293824, 231150385152, 392115499776, 440136281088, 326947332096, 155008401408, 42568187904, 5159780352;
%F Formula
The row polynomials satisfy P(n,x) = (n+6+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[(n + 6 + 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] = (n + 6 + 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

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.