Real-rooted atlas

A178822 → basis A013610 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-n + 3\,n x + 3\,n^{2}\right)\,P_{n}(x) = \left(20 + 60\,x + 45\,x^{2} + 34\,n + 57\,n x + 9\,n x^{2} + 6\,n^{2} + 9\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1301848
23366301891155
3246480647560151219600
414112665281088646804010206267750
5685444233609979201088640510300612363150000
629568022619526985440109771208981280336798033679833206250
711658241064448040715136838252809879408064665216202078801732104321750000
842831364546713620756736052929676881729648077059382442032390411258676084440072915859375
914862848179891712954809856290594304055576160646865290432539415676825219434245910804903940536625025000000
104920115266882816040475635201432214784032691859200500185445765148967824034676722080141859317602955402450177324147205283203125
111565491202361655296160518758406476101632017186577408031384184832040014835660835307207936020806033248075658302720141859317607737780961620937500000
124815749127984005120602222100482728818892808257029580801753030895616266765571072029153665981442250834505920117900855072038585734387265771138160328855690812388085937500
1314388756482600504524821556813824010839997808643683905505280891759194726415777278060544205790583398401967872453747213505007035520636664617388818942087790082959701217201366015946492039062500000
14419128934482015911936741143789568409579462656015446996877312419965227601928471712349900812847212135014414662579067136012463192207065676978540102464329908029010568997491700288129771514908055616363532667133789062500
Technical details
Node id
n-047ae4be85f661f5 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-bff12ef9596281bf

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:       30,        18;
  2:      336,       630,       189;
  3:     2464,      8064,      7560,       1512;
  4:    14112,     66528,    108864,      68040,      10206;
  5:    68544,    423360,    997920,    1088640,     510300,      61236;
  6:   295680,   2261952,   6985440,   10977120,    8981280,    3367980,     336798;
  7:  1165824,  10644480,  40715136,   83825280,   98794080,   64665216,   20207880,    1732104;
  8:  4283136,  45467136, 207567360,  529296768,  817296480,  770593824,  420323904,  112586760,   8444007;
  9: 14862848, 179891712, 954809856, 2905943040, 5557616064, 6865290432, 5394156768, 2521943424, 591080490, 39405366;
%F Formula
The row polynomials satisfy (3*n^2-n+3*n*x)*P(n,x) = (6*n^2+34*n+20+(9*n^2+57*n+60)*x+(9*n+45)*x^2)*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[((6*n^2 + 34*n + 20 + (9*n^2 + 57*n + 60)*x + (9*n + 45)*x^2)*P[n - 1, x])/(3*n^2 - n + 3*n*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^2 + 34*n + 20 + (9*n^2 + 57*n + 60)*x + (9*n + 45)*x^2)*v[i - 1])/(3*n^2 - n + 3*n*x)); 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.