Real-rooted atlas

A038314 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(25 + 150\,x + 144\,x^{2} + 90\,n x + 144\,n x^{2} + 36\,n^{2} x^{2}\right)\,P_{n}(x) = \left(250 + 2700\,x + 6120\,x^{2} + 3888\,x^{3} + 900\,n x + 3240\,n x^{2} + 2592\,n x^{3} + 360\,n^{2} x^{2} + 432\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 100 + 960\,x + 1296\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
010096012962356
1100014400388802764881928
21000019200077760011059205184002603920
31000002400000129600002764800025920000895795277985952
41000000288000001944000005529600007776000005374771201463132162238550336
51000000033600000027216000009676800000181440000001881169920010241925120229323571262235260032
6100000000384000000036288000000154828800000362880000000501645312000409677004800183458856960348285173761687546491136
71000000000432000000004665600000002322432000000653184000000011287019520000122903101440008255648563200313456656384051597803520044848554826240

All 15 rows

Technical details
Node id
n-d6ec43e2b2ecc097 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-dc876aec23f20348

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:          100,           960,           1296;
  1:         1000,         14400,          38880,           27648;
  2:        10000,        192000,         777600,         1105920,           518400;
  3:       100000,       2400000,       12960000,        27648000,         25920000,          8957952;
  4:      1000000,      28800000,      194400000,       552960000,        777600000,        537477120,        146313216;
  5:     10000000,     336000000,     2721600000,      9676800000,      18144000000,      18811699200,      10241925120,       2293235712;
  6:    100000000,    3840000000,    36288000000,    154828800000,     362880000000,     501645312000,     409677004800,     183458856960,      34828517376;
  7:   1000000000,   43200000000,   466560000000,   2322432000000,    6531840000000,   11287019520000,   12290310144000,    8255648563200,    3134566563840,     515978035200;
  8:  10000000000,  480000000000,  5832000000000,  33177600000000,  108864000000000,  225740390400000,  307257753600000,  275188285440000,  156728328192000,   51597803520000,   7492001071104;
  9: 100000000000, 5280000000000, 71280000000000, 456192000000000, 1710720000000000, 4138573824000000, 6759670579200000, 7567677849600000, 5746705367040000, 2837879193600000, 824120117821440, 106993205379072;
%F Formula
The row polynomials satisfy (25+(90*n+150)*x+(36*n^2+144*n+144)*x^2)*P(n,x) = (250+(900*n+2700)*x+(360*n^2+3240*n+6120)*x^2+(432*n^2+2592*n+3888)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 100 + 960*x + 1296*x^2.
%t Mathematica
P[0, x_] := 100 + 960*x + 1296*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((250 + (900*n + 2700)*x + (360*n^2 + 3240*n + 6120)*x^2 + (432*n^2 + 2592*n + 3888)*x^3)*P[n - 1, x])/(25 + (90*n + 150)*x + (36*n^2 + 144*n + 144)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 100 + 960*x + 1296*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((250 + (900*n + 2700)*x + (360*n^2 + 3240*n + 6120)*x^2 + (432*n^2 + 2592*n + 3888)*x^3)*v[i - 1])/(25 + (90*n + 150)*x + (36*n^2 + 144*n + 144)*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.