Real-rooted atlas

A038305 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(100 + 150\,x + 36\,x^{2} + 90\,n x + 36\,n x^{2} + 9\,n^{2} x^{2}\right)\,P_{n}(x) = \left(1000 + 2700\,x + 1530\,x^{2} + 243\,x^{3} + 900\,n x + 810\,n x^{2} + 162\,n x^{3} + 90\,n^{2} x^{2} + 27\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 100 + 240\,x + 81\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
010024081421
11000360024304327462
2100004800048600172802025125905
310000060000081000043200010125087482051998
41000000720000012150000864000030375005248803572132588101
5100000008400000017010000015120000070875000183708002500470139968507186238
6100000000960000000226800000024192000001417500000489888000100018800111974405314417766335681
7100000000010800000000291600000003628800000025515000000110224800003000564000503884800478296901968300117339726790
8100000000001200000000003645000000005184000000004252500000002204496000007501410000016796160000239148450019683000071449291753005319429
910000000000013200000000004455000000000712800000000066825000000004041576000000165031020000046189440000087687765000108256500007859421902550916825938605466358
101000000000000144000000000005346000000000095040000000000100237500000000692841600000003300620400000011085465600000263063295000043302600000047156531400306110016089813529380627295995089
111000000000000015600000000000063180000000000012355200000000001447875000000000112586760000000061297236000000024018508800000068396456700000140733450000002043449694000198971510400116757587703124873085544944259150478
1210000000000000016800000000000007371000000000000157248000000000002027025000000000017513496000000000107270163000000004803701760000000159591732300000039405366000000071520739290000928533715200081730311390043748223120107616802580261903246947045
131000000000000000180000000000000008505000000000000019656000000000000027641250000000000026270244000000000017878360500000000090069408000000000341982283500000009851341500000000214562217870000034820014320000040865155695000328111673400016142520375036733201921155165656542852942
14100000000000000001920000000000000009720000000000000002419200000000000000368550000000000000038211264000000000002860537680000000000160123392000000000068396456700000000022517352000000000057216591432000000111424045824000001634606227800000174992892480000129140163000005877312307201244050236916540918196322713089
Technical details
Node id
n-b618acefa39fed0e · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-09fd8ade2935dd5d

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,           240,            81;
  1:         1000,          3600,          2430,           432;
  2:        10000,         48000,         48600,         17280,          2025;
  3:       100000,        600000,        810000,        432000,        101250,          8748;
  4:      1000000,       7200000,      12150000,       8640000,       3037500,        524880,         35721;
  5:     10000000,      84000000,     170100000,     151200000,      70875000,      18370800,       2500470,       139968;
  6:    100000000,     960000000,    2268000000,    2419200000,    1417500000,     489888000,     100018800,     11197440,      531441;
  7:   1000000000,   10800000000,   29160000000,   36288000000,   25515000000,   11022480000,    3000564000,    503884800,    47829690,     1968300;
  8:  10000000000,  120000000000,  364500000000,  518400000000,  425250000000,  220449600000,   75014100000,  16796160000,  2391484500,   196830000,   7144929;
  9: 100000000000, 1320000000000, 4455000000000, 7128000000000, 6682500000000, 4041576000000, 1650310200000, 461894400000, 87687765000, 10825650000, 785942190, 25509168;
%F Formula
The row polynomials satisfy (100+(90*n+150)*x+(9*n^2+36*n+36)*x^2)*P(n,x) = (1000+(900*n+2700)*x+(90*n^2+810*n+1530)*x^2+(27*n^2+162*n+243)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 100 + 240*x + 81*x^2.
%t Mathematica
P[0, x_] := 100 + 240*x + 81*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((1000 + (900*n + 2700)*x + (90*n^2 + 810*n + 1530)*x^2 + (27*n^2 + 162*n + 243)*x^3)*P[n - 1, x])/(100 + (90*n + 150)*x + (9*n^2 + 36*n + 36)*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 + 240*x + 81*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((1000 + (900*n + 2700)*x + (90*n^2 + 810*n + 1530)*x^2 + (27*n^2 + 162*n + 243)*x^3)*v[i - 1])/(100 + (90*n + 150)*x + (9*n^2 + 36*n + 36)*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.