Real-rooted atlas

A156992 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4 - 6\,x + 18\,n x + 9\,n^{2} x^{2}\right)\,P_{n}(x) = \left(8 + 36\,x + 54\,x^{2} + 27\,x^{3} + 8\,n + 72\,n x + 144\,n x^{2} + 81\,n x^{3} + 36\,n^{2} x + 108\,n^{2} x^{2} + 81\,n^{2} x^{3} + 18\,n^{3} x^{2} + 27\,n^{3} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
142428
224288486798
31923456116641036825680
4192046080233280414720243000939000
52304069120046656001244160014580000629856038700000
632256011612160979776003483648006123600005290790401800338401779750000
7516096021676032021946982409754214400228614400002962842624020163790080564350976090468000000
892897280445906944052672757760280921374720823011840000142216445952014517928857608126654054401928493100805040630000000
918579456001003290624001354442342400842764124160029628426240000639974006784008710757314560073139886489600347128758144007142567040000305613000000000
104087480320024524881920003724716441600026486872473600010863756288000002815885629849600479091652300800053635916759040003818416339584000157136474880000028520270190720020036362500000000
1198099527680064745688268800109258348953600087406679162880004097188085760000012389896771338240025296039241482240035399705060966400033602063788339200020742014684160000075293513303500800122189322866688001412680500000000000
12255058771968001836423158169600340886048735232003030098210979840001597903353446400000552235398951075840013153940405570764800220894159580430336002620960975490457600021571695271526400000117457880753461248003812306873440665600559270713204403200106601244750000000000
1371416456151040055704835797811200112802292490567680011029557487966617600646263134060544000002512671065227395072006840049010896797696001340091234787944038400190805959015705313280019630242697089024000001425155619808663142400693839850966201139200203574539606402764800272421095331280896008573099535000000000000
142142493684531200017996946950062080003948080237169868800042112855863145267200027143051630542848000001172579830439451033600035910257307208187904000804054740872766423040001335641713109937192960001648940386555478016000001496413400799096299520009713757913526815948800042750653317344580608000114416860039137976320001407277341953683200000732055011187500000000000
Technical details
Node id
n-ffcf6e8750396c7d · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-56805ed74e931bf6

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:          4,           24;
  2:         24,          288,           486;
  3:        192,         3456,         11664,         10368;
  4:       1920,        46080,        233280,        414720,         243000;
  5:      23040,       691200,       4665600,      12441600,       14580000,        6298560;
  6:     322560,     11612160,      97977600,     348364800,      612360000,      529079040,      180033840;
  7:    5160960,    216760320,    2194698240,    9754214400,    22861440000,    29628426240,    20163790080,     5643509760;
  8:   92897280,   4459069440,   52672757760,  280921374720,   823011840000,  1422164459520,  1451792885760,   812665405440,   192849310080;
  9: 1857945600, 100329062400, 1354442342400, 8427641241600, 29628426240000, 63997400678400, 87107573145600, 73139886489600, 34712875814400, 7142567040000;
%F Formula
The row polynomials satisfy (4+(18*n-6)*x+9*n^2*x^2)*P(n,x) = (8*n+8+(36*n^2+72*n+36)*x+(18*n^3+108*n^2+144*n+54)*x^2+(27*n^3+81*n^2+81*n+27)*x^3)*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[Together[((8*n + 8 + (36*n^2 + 72*n + 36)*x + (18*n^3 + 108*n^2 + 144*n + 54)*x^2 + (27*n^3 + 81*n^2 + 81*n + 27)*x^3)*P[n - 1, x])/(4 + (18*n - 6)*x + 9*n^2*x^2)]];
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] = ((8*n + 8 + (36*n^2 + 72*n + 36)*x + (18*n^3 + 108*n^2 + 144*n + 54)*x^2 + (27*n^3 + 81*n^2 + 81*n + 27)*x^3)*v[i - 1])/(4 + (18*n - 6)*x + 9*n^2*x^2)); 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.