Real-rooted atlas

A038329 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(25 + 100\,x + 64\,x^{2} + 60\,n x + 64\,n x^{2} + 16\,n^{2} x^{2}\right)\,P_{n}(x) = \left(375 + 2700\,x + 4080\,x^{2} + 1728\,x^{3} + 900\,n x + 2160\,n x^{2} + 1152\,n x^{3} + 240\,n^{2} x^{2} + 192\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 225 + 1440\,x + 1296\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0225144012962961
13375324005832027648121743
250625648000174960016588805184004625505
3759375121500004374000062208000388800008957952166695327
411390625218700000984150000186624000017496000008062156801463132165782609521
5170859375382725000020667150000489888000006123600000042326323200153628876802293235712194872505967
6256289062565610000000413343000000117573120000018370800000001693052928000921773260800275188285440348285173766419170082241
7384433593751107168750000797161500000026453952000000496011600000005714053632000041479796736000185752092672004701849845760515978035200207585709313535

All 15 rows

Technical details
Node id
n-e782416b20ea362c · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-7f34919e5b06c424

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:           225,            1440,             1296;
  1:          3375,           32400,            58320,             27648;
  2:         50625,          648000,          1749600,           1658880,            518400;
  3:        759375,        12150000,         43740000,          62208000,          38880000,           8957952;
  4:      11390625,       218700000,        984150000,        1866240000,        1749600000,         806215680,         146313216;
  5:     170859375,      3827250000,      20667150000,       48988800000,       61236000000,       42326323200,       15362887680,        2293235712;
  6:    2562890625,     65610000000,     413343000000,     1175731200000,     1837080000000,     1693052928000,      921773260800,      275188285440,       34828517376;
  7:   38443359375,   1107168750000,    7971615000000,    26453952000000,    49601160000000,    57140536320000,    41479796736000,    18575209267200,     4701849845760,     515978035200;
  8:  576650390625,  18452812500000,  149467781250000,   566870400000000,  1240029000000000,  1714216089600000,  1555492377600000,   928760463360000,   352638738432000,   77396705280000,    7492001071104;
  9: 8649755859375, 304471406250000, 2740242656250000, 11691702000000000, 29229255000000000, 47140942464000000, 51331248460800000, 38311369113600000, 19395130613760000, 6385228185600000, 1236180176732160, 106993205379072;
%F Formula
The row polynomials satisfy (25+(60*n+100)*x+(16*n^2+64*n+64)*x^2)*P(n,x) = (375+(900*n+2700)*x+(240*n^2+2160*n+4080)*x^2+(192*n^2+1152*n+1728)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 225 + 1440*x + 1296*x^2.
%t Mathematica
P[0, x_] := 225 + 1440*x + 1296*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((375 + (900*n + 2700)*x + (240*n^2 + 2160*n + 4080)*x^2 + (192*n^2 + 1152*n + 1728)*x^3)*P[n - 1, x])/(25 + (60*n + 100)*x + (16*n^2 + 64*n + 64)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 225 + 1440*x + 1296*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((375 + (900*n + 2700)*x + (240*n^2 + 2160*n + 4080)*x^2 + (192*n^2 + 1152*n + 1728)*x^3)*v[i - 1])/(25 + (60*n + 100)*x + (16*n^2 + 64*n + 64)*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.