Real-rooted atlas

A038336 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(121 + 550\,x + 400\,x^{2} + 330\,n x + 400\,n x^{2} + 100\,n^{2} x^{2}\right)\,P_{n}(x) = \left(2662 + 21780\,x + 37400\,x^{2} + 18000\,x^{3} + 7260\,n x + 19800\,n x^{2} + 12000\,n x^{3} + 2200\,n^{2} x^{2} + 2000\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 484 + 3520\,x + 3600\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0484352036007604
110648116160237600128000492408
223425634073601045440011264000400000029360016
35153632937024003833280006195200004400000001152000001656904032
4113379904247374336012649824000272588800002904000000015206400000313600000089878227264
5249435788863492746240389614579200104946688000014907200000001170892800000482944000000819200000004731545363328
65487587353615963890483201142869432320036941234176000655916800000006869237760000042499072000000144179200000002073600000000243295843021056
7120726921779239510628945920323268782284800121906072780800025974305280000003400272691200000280493875200000014273740800000004105728000000005120000000000012274836259456512

All 15 rows

Technical details
Node id
n-122a5becb4c16dcb · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-4b3bb4c5c4994d07

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:             484,              3520,               3600;
  1:           10648,            116160,             237600,              128000;
  2:          234256,           3407360,           10454400,            11264000,             4000000;
  3:         5153632,          93702400,          383328000,           619520000,           440000000,           115200000;
  4:       113379904,        2473743360,        12649824000,         27258880000,         29040000000,         15206400000,          3136000000;
  5:      2494357888,       63492746240,       389614579200,       1049466880000,       1490720000000,       1170892800000,        482944000000,         81920000000;
  6:     54875873536,     1596389048320,     11428694323200,      36941234176000,      65591680000000,      68692377600000,      42499072000000,      14417920000000,       2073600000000;
  7:   1207269217792,    39510628945920,    323268782284800,    1219060727808000,    2597430528000000,    3400272691200000,    2804938752000000,    1427374080000000,     410572800000000,      51200000000000;
  8:  26559922791424,   965815374233600,   8889891512832000,   38313337159680000,   95239119360000000,  149611998412800000,  154271631360000000,  104674099200000000,   45163008000000000,   11264000000000000,   1239040000000000;
  9: 584318301411328, 23372732056453120, 239039305122816000, 1158978449080320000, 3292552412160000000, 6034350602649600000, 7466746957824000000, 6332783001600000000, 3643149312000000000, 1362944000000000000, 299847680000000000, 29491200000000000;
%F Formula
The row polynomials satisfy (121+(330*n+550)*x+(100*n^2+400*n+400)*x^2)*P(n,x) = (2662+(7260*n+21780)*x+(2200*n^2+19800*n+37400)*x^2+(2000*n^2+12000*n+18000)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 484 + 3520*x + 3600*x^2.
%t Mathematica
P[0, x_] := 484 + 3520*x + 3600*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2662 + (7260*n + 21780)*x + (2200*n^2 + 19800*n + 37400)*x^2 + (2000*n^2 + 12000*n + 18000)*x^3)*P[n - 1, x])/(121 + (330*n + 550)*x + (100*n^2 + 400*n + 400)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 484 + 3520*x + 3600*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((2662 + (7260*n + 21780)*x + (2200*n^2 + 19800*n + 37400)*x^2 + (2000*n^2 + 12000*n + 18000)*x^3)*v[i - 1])/(121 + (330*n + 550)*x + (100*n^2 + 400*n + 400)*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.