Real-rooted atlas

A088996 → basis A007318 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-9\,n - 6\,n x - n x^{2} + 12\,n^{2} + 10\,n^{2} x + 2\,n^{2} x^{2} - 4\,n^{3} - 4\,n^{3} x - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(1 - 6\,n - 2\,n x + 4\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1213
289219
34984436182
4403920737238242322
5415211785127886537152612036908
6513421737892371121656336156411196720701356
7740211290528047639264211292215005162462892772504015493200
812184245543414961039019461109176527195446928807236684854085809640320389854000
92253753901124878617245423578830661470222409692202122918980540358839681044316877118436288011003285600
10462672666025535607273625490794028942333283882454944228510690482852140580725459457480041032148296982353603628800344144306400
1110436327968563083371942817130222007292755743203066291367132029921211631117361081289953451383961074994922991848761409799477611967105603991680011811681670400
1225653471596551684890253238450210833751811897100093815461069076760872978937875512550853627961208345231881762381707133656755580151010494447220579178931215757614720479001600441303451590400
1368244722853300483810545598705156920520856996830814384756188074083992704055894385321448911685926581191793047801354248924983297507795868457666137682314695516260200801126323199358370816223051052160622702080017827201149811200
1419532894671678501486305502092868552106190004053216111502568823586455162609262067363482170807604554456003133105901914884708780386733825655533450737352962520811412313769286840276427489794195247158205144462452802792544384337802258304087178291200774198357476723200
Technical details
Node id
n-f47afb12e1401479 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-de4ba04a7fe52d87

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, 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:         2,          1;
  2:         8,          9,          2;
  3:        49,         84,         43,          6;
  4:       403,        920,        737,        238,         24;
  5:      4152,      11785,      12788,       6537,       1526,        120;
  6:     51342,     173789,     237112,     165633,      61564,      11196,       720;
  7:    740211,    2905280,    4763926,    4211292,    2150051,     624628,     92772,     5040;
  8:  12184245,   54341496,  103901946,  110917652,   71954469,   28807236,   6848540,   858096,   40320;
  9: 225375390, 1124878617, 2454235788, 3066147022, 2409692202, 1229189805, 403588396, 81044316, 8771184, 362880;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(4*n^3-12*n^2+9*n+(4*n^3-10*n^2+6*n)*x+(n^3-2*n^2+n)*x^2)*P(n-2,x) + (4*n^2-6*n+1+(2*n^2-2*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 - 12*n^2 + 9*n + (4*n^3 - 10*n^2 + 6*n)*x + (n^3 - 2*n^2 + n)*x^2)*P[n - 2, x] + (4*n^2 - 6*n + 1 + (2*n^2 - 2*n)*x)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^3 - 12*n^2 + 9*n + (4*n^3 - 10*n^2 + 6*n)*x + (n^3 - 2*n^2 + n)*x^2)*v[i - 2] + (4*n^2 - 6*n + 1 + (2*n^2 - 2*n)*x)*v[i - 1])/(n - 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

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.