Real-rooted atlas

A038284 → basis A007318 → x ↦ x+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(20 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 400 + 240\,x + 36\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
040024036676
180007200216021617576
216000019200086400172801296456976
3320000048000002880000864000129600777611881376
4640000001152000008640000034560000777600093312046656308915776
512800000002688000000241920000012096000003628800006531840065318402799368031810176
625600000000614400000006451200000038707200000145152000003483648000522547200447897601679616208827064576
7512000000000138240000000016588800000001161216000000522547200000156764160000313528320004031078400302330880100776965429503678976
810240000000000307200000000004147200000000033177600000000174182400000006270566400000156764160000026873856000030233088000201553920060466176141167095653376
9204800000000000675840000000000101376000000000091238400000000054743040000000022992076800000068976230400000147806208000002217093120000221709312000133025587203627970563670344486987776
104096000000000000147456000000000002433024000000000024330240000000000164229120000000007882997760000000275904921600000070946979840000013302558720000017736744960000159630704640087071293440217678233695428956661682176
1181920000000000000319488000000000000575078400000000000632586240000000000474439680000000000256197427200000000102478970880000000307436912640000006917330534400000115288842240000013834661068800011319268147200565963407360130606940162481152873203736576
121638400000000000000688128000000000000013418496000000000000161021952000000000001328431104000000000079705866240000000003586763980800000000122974765056000000032280875827200000064561751654400000968426274816000010564650270720007923487703040036569943244807836416409664509974703297150976
133276800000000000000014745600000000000000030965760000000000000040255488000000000000036229939200000000000023911759872000000000011955879936000000000046115536896000000000138346610688000000003228087582720000000581055764889600000792348770304000007923487703040000548549148672000235092492288004701849845761677259342285725925376
146553600000000000000003145728000000000000000707788800000000000000099090432000000000000009661317120000000000000695614832640000000000038258815795200000000001639663534080000000000553386442752000000000147569718067200000000309896407941120000005071032129945600000633879016243200000585119091916800003761479876608000150459195064320282110990745643608742899428874059776
Technical details
Node id
n-4e07e76af0d9dd55 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-1024c0779b1dc97d

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:             400,             240,               36;
  1:            8000,            7200,             2160,             216;
  2:          160000,          192000,            86400,           17280,            1296;
  3:         3200000,         4800000,          2880000,          864000,          129600,            7776;
  4:        64000000,       115200000,         86400000,        34560000,         7776000,          933120,          46656;
  5:      1280000000,      2688000000,       2419200000,      1209600000,       362880000,        65318400,        6531840,         279936;
  6:     25600000000,     61440000000,      64512000000,     38707200000,     14515200000,      3483648000,      522547200,       44789760,       1679616;
  7:    512000000000,   1382400000000,    1658880000000,   1161216000000,    522547200000,    156764160000,    31352832000,     4031078400,     302330880,     10077696;
  8:  10240000000000,  30720000000000,   41472000000000,  33177600000000,  17418240000000,   6270566400000,  1567641600000,   268738560000,   30233088000,   2015539200,    60466176;
  9: 204800000000000, 675840000000000, 1013760000000000, 912384000000000, 547430400000000, 229920768000000, 68976230400000, 14780620800000, 2217093120000, 221709312000, 13302558720, 362797056;
%F Formula
The row polynomials satisfy P(n,x) = (20+6*x)*P(n-1,x) for n >= 1, with P(0,x) = 400 + 240*x + 36*x^2.
%t Mathematica
P[0, x_] := 400 + 240*x + 36*x^2;
P[n_, x_] := P[n, x] = Expand[(20 + 6*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 400 + 240*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = (20 + 6*x)*v[i - 1]); 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.