Real-rooted atlas

A038301 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(100 + 550\,x + 484\,x^{2} + 330\,n x + 484\,n x^{2} + 121\,n^{2} x^{2}\right)\,P_{n}(x) = \left(2000 + 19800\,x + 41140\,x^{2} + 23958\,x^{3} + 6600\,n x + 21780\,n x^{2} + 15972\,n x^{3} + 2420\,n^{2} x^{2} + 2662\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 400 + 3520\,x + 4356\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0400352043568276
18000105600261360170368545328
216000028160001045440013629440585640032916240
33200000704000003484800006814720005856400001855307521874722752
4640000001689600000104544000002725888000035138400000222636902405555615296102424585536
5128000000039424000000292723200000954060800000163979200000015584583168007777861414401596389048325423163363072
625600000000901120000000780595200000030529945600000655916800000008311777689600062222891315200255422247731204444945756416280182136340736
751200000000020275200000000200724480000000915898368000000236130048000000037402999603200003733373478912000229880022958080080009023615488012072692177920014192001354746880

All 15 rows

Technical details
Node id
n-3a464fe5de3ea776 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-d22498ba92effd6b

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:             400,             3520,               4356;
  1:            8000,           105600,             261360,             170368;
  2:          160000,          2816000,           10454400,           13629440,             5856400;
  3:         3200000,         70400000,          348480000,          681472000,           585640000,           185530752;
  4:        64000000,       1689600000,        10454400000,        27258880000,         35138400000,         22263690240,          5555615296;
  5:      1280000000,      39424000000,       292723200000,       954060800000,       1639792000000,       1558458316800,        777786141440,        159638904832;
  6:     25600000000,     901120000000,      7805952000000,     30529945600000,      65591680000000,      83117776896000,      62222891315200,      25542224773120,       4444945756416;
  7:    512000000000,   20275200000000,    200724480000000,    915898368000000,    2361300480000000,    3740299960320000,    3733373478912000,    2298800229580800,     800090236154880,     120726921779200;
  8:  10240000000000,  450560000000000,   5018112000000000,  26168524800000000,   78710016000000000,  149611998412800000,  186668673945600000,  153253348638720000,   80009023615488000,   24145384355840000,   3213750657762304;
  9: 204800000000000, 9912320000000000, 122664960000000000, 719634432000000000, 2473743360000000000, 5485773275136000000, 8213421653606400000, 8428934175129600000, 5867328398469120000, 2655992279142400000, 707025144707706880, 84141835403231232;
%F Formula
The row polynomials satisfy (100+(330*n+550)*x+(121*n^2+484*n+484)*x^2)*P(n,x) = (2000+(6600*n+19800)*x+(2420*n^2+21780*n+41140)*x^2+(2662*n^2+15972*n+23958)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 400 + 3520*x + 4356*x^2.
%t Mathematica
P[0, x_] := 400 + 3520*x + 4356*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2000 + (6600*n + 19800)*x + (2420*n^2 + 21780*n + 41140)*x^2 + (2662*n^2 + 15972*n + 23958)*x^3)*P[n - 1, x])/(100 + (330*n + 550)*x + (121*n^2 + 484*n + 484)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 400 + 3520*x + 4356*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((2000 + (6600*n + 19800)*x + (2420*n^2 + 21780*n + 41140)*x^2 + (2662*n^2 + 15972*n + 23958)*x^3)*v[i - 1])/(100 + (330*n + 550)*x + (121*n^2 + 484*n + 484)*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

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.