Real-rooted atlas

A088996 → basis A013611 → x ↦ x+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(-1 + 6\,n + 4\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1549
2558432171
3935208815523844959
4215056298469104336646144193401
56236452256636326369623583368514561228809476649
6218275759394974016838862416087046486401024247357442949120559122291
78949305754463111440953452630411310570496804681497634333941768135147528257536038579438079
8420617370252384040160805909423023688367016550407401385597444188676055041481038069762991351398426424115203047775608241
922292720623251414963538514039902486604384656191106023686934860324787248844971242496229287355678726917153292288121693431398495126814720271252029133449
1013152705167717592399937021626029202321250642565467626989715072671633201571916856557974332221443306594248204288132526146695987234848525621657654289854627840380507258880026853950884211451
11854925835901637565847149337852600230471060418692080483885967834306944677137168569209088663145441851082752463783713195053056231632142273380352809630462003118081886219182433894426360833218969601674231939072002927080646379048159
12606997343490162625508551200422032060019524108457944062480454185146163330128647130326542446057856079585910462748911616647596248457007390723870750033477255168016866719109344395264522544183714303180810925471234229534721384190462184652808036313307545600348322596919106730921
13467387954487425221254231482861864531427001767802975481149482160451247926527073323948878521141977327812180489835884907703616279552912495845718190031257663479780177539249111043311527388827691122688127942841514592200294435584910403500507136067470716976815407104781658652804120576041788829199237120044933615002564768288809
14387932002224562933637537738680298605192083100170423950991376783010480473532745644415229878464904424317266981902509312125609684241243589941350413081811067775002375987201037879849075762185125888630343542266918158204928291638092231462229114880101183466883077053087744255276193350359627857924427136832529082875904472413573805674332160234017443515727872006245772485356502792144451
Technical details
Node id
n-ddbd7b5c47ccc909 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-82d3622bce92de08

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:             5,              4;
  2:            55,             84,             32;
  3:           935,           2088,           1552,            384;
  4:         21505,          62984,          69104,          33664,           6144;
  5:        623645,        2256636,        3263696,        2358336,         851456,         122880;
  6:      21827575,       93949740,      168388624,      160870464,       86401024,       24735744,        2949120;
  7:     894930575,     4463111440,     9534526304,    11310570496,     8046814976,     3433394176,      813514752,      82575360;
  8:   42061737025,   238404016080,   590942302368,   836701655040,   740138559744,   418867605504,   148103806976,   29913513984,    2642411520;
  9: 2229272062325, 14149635385140, 39902486604384, 65619110602368, 69348603247872, 48844971242496, 22928735567872, 6917153292288, 1216934313984, 95126814720;
%F Formula
The row polynomials satisfy P(n,x) = (6*n-1+4*n*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[(6*n - 1 + 4*n*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] = (6*n - 1 + 4*n*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. [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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.