Real-rooted atlas

A085881 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 2\,x + 6\,n x + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(-2 - 12\,x - 24\,x^{2} - 16\,x^{3} + 4\,n + 12\,n x + 8\,n x^{2} + 24\,n^{2} x + 72\,n^{2} x^{2} + 48\,n^{2} x^{3} + 16\,n^{3} x^{2} + 32\,n^{3} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
121618
212192432636
31202880129601536031320
41680537603628808601606720001950480
530240120960010886400387072006048000034836480146149920
666528031933440359251200170311680039916800004598415360208631808012771380160
71729728096864768013076743680774918144002421619200004184557977603797098905601416993177601273581429120
85189184003321077760052306974720037196070912001452971520000033476463820800455651868672003400783626240010760291942400142615900627200
917643225600127031224320022865620377600189699961651200889218570240000256094948229120046476490604544005203198948147200329264933437440090333315072000017710852083724800
106704425728005363540582400010861169679360001029799791820800056317176115200000194632160654131200441526660743168000659071866765312000625603373531136000343266597273600000830705165402112002414926772357299200
112815858805760024779557490688005575400435404800059470937977651200037169336236032000001498667637036810240040797063452668723200761228006113935360009634291952379494400079294583970201600000383785786415775744008304305521242931200358596831213318297600
121295295050649600124348324862361600307762104034344960036475508626292736000256468420028620800000118180647949188464640037533298376455225344008403957187497846374400132953228942837022720001459020345051709440000010592487705075410534400458397664772609802240089663432114086871040057593865716693597491200
136476475253248000067355342633779200001818594251112038400002370908060709027840000185227192242892800000009602177645871562752000034852348492422709248000091042869531226669056000017283919762568812953600002370908060709027840000000229503900276633894912000014897924105109818572800005828123087415646617600001039883591382309273600009946988361827360363520000
1434972966367539200003916972233164390400001145714378200584192000016294604489963864064000014003175733562695680000008065829222532112711680000329354693253394602393600009832629909372480258048000021777738900836704321536000035848129877920500940800000043376237152283806138368000037542768744876742803456000022030305270431144214528000078615199508502581084160000128924343217296506880000001838162146891875458734080000
Technical details
Node id
n-d73b68d236fd0cef · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-181bf1738e48841f

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:           2,            16;
  2:          12,           192,            432;
  3:         120,          2880,          12960,           15360;
  4:        1680,         53760,         362880,          860160,          672000;
  5:       30240,       1209600,       10886400,        38707200,        60480000,         34836480;
  6:      665280,      31933440,      359251200,      1703116800,      3991680000,       4598415360,       2086318080;
  7:    17297280,     968647680,    13076743680,     77491814400,    242161920000,     418455797760,     379709890560,     141699317760;
  8:   518918400,   33210777600,   523069747200,   3719607091200,  14529715200000,   33476463820800,   45565186867200,   34007836262400,   10760291942400;
  9: 17643225600, 1270312243200, 22865620377600, 189699961651200, 889218570240000, 2560949482291200, 4647649060454400, 5203198948147200, 3292649334374400, 903333150720000;
%F Formula
The row polynomials satisfy (1+(6*n-2)*x+4*n^2*x^2)*P(n,x) = (4*n-2+(24*n^2+12*n-12)*x+(16*n^3+72*n^2+8*n-24)*x^2+(32*n^3+48*n^2-16)*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[((4*n - 2 + (24*n^2 + 12*n - 12)*x + (16*n^3 + 72*n^2 + 8*n - 24)*x^2 + (32*n^3 + 48*n^2 - 16)*x^3)*P[n - 1, x])/(1 + (6*n - 2)*x + 4*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] = ((4*n - 2 + (24*n^2 + 12*n - 12)*x + (16*n^3 + 72*n^2 + 8*n - 24)*x^2 + (32*n^3 + 48*n^2 - 16)*x^3)*v[i - 1])/(1 + (6*n - 2)*x + 4*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.