Real-rooted atlas

A200536 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(36 - 42\,x - 72\,x^{2} - 14\,x^{3} + 4\,x^{4} + 126\,n x + 96\,n x^{2} - 14\,n x^{3} - 16\,n x^{4} + 49\,n^{2} x^{2} + 56\,n^{2} x^{3} + 16\,n^{2} x^{4}\right)\,P_{n}(x) = \left(216 + 756\,x + 1098\,x^{2} + 847\,x^{3} + 366\,x^{4} + 84\,x^{5} + 8\,x^{6} + 756\,n x + 2046\,n x^{2} + 2198\,n x^{3} + 1170\,n x^{4} + 308\,n x^{5} + 32\,n x^{6} + 294\,n^{2} x^{2} + 679\,n^{2} x^{3} + 586\,n^{2} x^{4} + 224\,n^{2} x^{5} + 32\,n^{2} x^{6}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
16281852
2363366574481001577
321630249882135529150302439239240
412962419211080822848025802517136067032143361296876825
5777618144010692002943360466275046572123046330130816035640056000387218292500
646656130636894128483241728066109500881889128092884152260096237994207504000156235219353610816363740625
7279936914457677892192322092288803514600134139499215849666421367041984873348966414010800144037432357880326019104614656288006980175000

All 15 rows

Technical details
Node id
n-e76af746ebfe4ce3 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-b81b61d2759417d6

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, intercept 0)
  • 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:        1;
  1:        6,        28,         18;
  2:       36,       336,        657,         448,         100;
  3:      216,      3024,       9882,       13552,        9150,         3024,          392;
  4:     1296,     24192,     110808,      228480,      258025,       171360,        67032,        14336,         1296;
  5:     7776,    181440,    1069200,     2943360,     4662750,      4657212,      3046330,      1308160,       356400,        56000,         3872;
  6:    46656,   1306368,    9412848,    32417280,    66109500,     88188912,     80928841,     52260096,     23799420,      7504000,      1562352,       193536,       10816;
  7:   279936,   9144576,   77892192,   322092288,   803514600,   1341394992,   1584966642,   1367041984,    873348966,    414010800,    144037432,     35788032,     6019104,      614656,      28800;
  8:  1679616,  62705664,  616419072,  2975035392,  8776706400,  17597212416,  25440090480,  27462779904,  22591325169,  14303531200,   6980160880,   2606994432,   732475744,   149976064,   21139200,   1835008,    73984;
  9: 10077696, 423263232, 4716361728, 26022850560, 88624238400, 207440134272, 354707561376, 459939658752, 462959070966, 366876048700, 230526944802, 114984914688, 45310338528, 13943164032, 3282379200, 571146240, 69249024, 5225472, 184832;
%F Formula
The row polynomials satisfy (36+(126*n-42)*x+(49*n^2+96*n-72)*x^2+(56*n^2-14*n-14)*x^3+(16*n^2-16*n+4)*x^4)*P(n,x) = (216+(756*n+756)*x+(294*n^2+2046*n+1098)*x^2+(679*n^2+2198*n+847)*x^3+(586*n^2+1170*n+366)*x^4+(224*n^2+308*n+84)*x^5+(32*n^2+32*n+8)*x^6)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((216 + (756*n + 756)*x + (294*n^2 + 2046*n + 1098)*x^2 + (679*n^2 + 2198*n + 847)*x^3 + (586*n^2 + 1170*n + 366)*x^4 + (224*n^2 + 308*n + 84)*x^5 + (32*n^2 + 32*n + 8)*x^6)*P[n - 1, x])/(36 + (126*n - 42)*x + (49*n^2 + 96*n - 72)*x^2 + (56*n^2 - 14*n - 14)*x^3 + (16*n^2 - 16*n + 4)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((216 + (756*n + 756)*x + (294*n^2 + 2046*n + 1098)*x^2 + (679*n^2 + 2198*n + 847)*x^3 + (586*n^2 + 1170*n + 366)*x^4 + (224*n^2 + 308*n + 84)*x^5 + (32*n^2 + 32*n + 8)*x^6)*v[i - 1])/(36 + (126*n - 42)*x + (49*n^2 + 96*n - 72)*x^2 + (56*n^2 - 14*n - 14)*x^3 + (16*n^2 - 16*n + 4)*x^4)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.