Real-rooted atlas

A038319 → basis A013609 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(13 + 5\,x + 5\,n\right)\,P_{n}(x) = \left(288 + 260\,x + 50\,x^{2} + 80\,n + 50\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 576 + 520\,x + 100\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
05765201001196
111776172807800100037856
2229376471040345600104000100001160016
3432537611468800117760005760000130000010000034730176
479691776259522560344064000235520000864000001560000010000001021798336
51442840576557842432090832896008028160000412160000012096000001820000001000000029655914496
62576980377611542724608022313697280024222105600016056320000065945600000161280000002080000000100000000851371878656
74552665333762319282339840519422607360066941091840005449973760000289013760000098918400000020736000000023400000000100000000024223939490816

All 15 rows

Technical details
Node id
n-dd032736ed5eddee · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-354a4e768d29be07

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:             576,             520,              100;
  1:           11776,           17280,             7800,             1000;
  2:          229376,          471040,           345600,           104000,            10000;
  3:         4325376,        11468800,         11776000,          5760000,          1300000,           100000;
  4:        79691776,       259522560,        344064000,        235520000,         86400000,         15600000,          1000000;
  5:      1442840576,      5578424320,       9083289600,       8028160000,       4121600000,       1209600000,        182000000,        10000000;
  6:     25769803776,    115427246080,     223136972800,     242221056000,     160563200000,      65945600000,      16128000000,      2080000000,       100000000;
  7:    455266533376,   2319282339840,    5194226073600,    6694109184000,    5449973760000,    2890137600000,     989184000000,    207360000000,     23400000000,     1000000000;
  8:   7971459301376,  45526653337600,  115964116992000,  173140869120000,  167352729600000,  108999475200000,   48168960000000,  14131200000000,   2592000000000,   260000000000,   10000000000;
  9: 138538465099776, 876860523151360, 2503965933568000, 4252017623040000, 4761373900800000, 3681760051200000, 1998323712000000, 756940800000000, 194304000000000, 31680000000000, 2860000000000, 100000000000;
%F Formula
The row polynomials satisfy (5*n+13+5*x)*P(n,x) = (80*n+288+(50*n+260)*x+50*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 576 + 520*x + 100*x^2.
%t Mathematica
P[0, x_] := 576 + 520*x + 100*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((80*n + 288 + (50*n + 260)*x + 50*x^2)*P[n - 1, x])/(5*n + 13 + 5*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 576 + 520*x + 100*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((80*n + 288 + (50*n + 260)*x + 50*x^2)*v[i - 1])/(5*n + 13 + 5*x)); 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.