Real-rooted atlas

basis-map coefficients of A297191, shifted → (θ+1)² → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 - 4\,n + 4\,n^{2}\right)\,P_{n}(x) = \left(1 + 6\,x + 9\,x^{2} + 4\,n + 24\,n x + 36\,n x^{2} + 4\,n^{2} + 24\,n^{2} x + 36\,n^{2} x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
195481144
2253001350270020256400
349882661526460595357144235721200704
4811944204121224724592701102248165337214171765314415308416
512136304900539204020582107409556185238903175524035724645238164307144929126877696
616960841003861003860677605532525064113837724292725576548860455731813940658632546359254116898135292835349504
722594501842752211300182432251094593504925670751688801400443310367588662073501329931102514508339300108812544755022117450107616802560397977600

All 15 rows

Technical details
Node id
n-c5a2bf19e77949e1 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-12929af9fbcc0757

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:   1;
  1:   9,    54,     81;
  2:  25,   300,   1350,    2700,     2025;
  3:  49,   882,   6615,   26460,    59535,     71442,      35721;
  4:  81,  1944,  20412,  122472,   459270,   1102248,    1653372,     1417176,       531441;
  5: 121,  3630,  49005,  392040,  2058210,   7409556,   18523890,    31755240,     35724645,     23816430,      7144929;
  6: 169,  6084, 100386, 1003860,  6776055,  32525064,  113837724,   292725576,    548860455,    731813940,    658632546,     359254116,      89813529;
  7: 225,  9450, 184275, 2211300, 18243225, 109459350,  492567075,  1688801400,   4433103675,   8866207350,  13299311025,   14508339300,   10881254475,    5022117450,    1076168025;
  8: 289, 13872, 312120, 4369680, 42604380, 306751536, 1687133448,  7230571920,  24403180230,  65075147280, 136657809288,  223621869744,  279527337180,  258025234320,  165873364920,   66349345968,   12440502369;
  9: 361, 19494, 497097, 7953552, 89477460, 751610664, 4885469316, 25125270768, 103641741918, 345472473060, 932775677262, 2035146932208, 3561507131364, 4931317566504, 5283554535540, 4226843628432, 2377599540993, 839152779174, 139858796529;
%F Formula
The row polynomials satisfy (4*n^2-4*n+1)*P(n,x) = (4*n^2+4*n+1+(24*n^2+24*n+6)*x+(36*n^2+36*n+9)*x^2)*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[((4*n^2 + 4*n + 1 + (24*n^2 + 24*n + 6)*x + (36*n^2 + 36*n + 9)*x^2)*P[n - 1, x])/(4*n^2 - 4*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; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^2 + 4*n + 1 + (24*n^2 + 24*n + 6)*x + (36*n^2 + 36*n + 9)*x^2)*v[i - 1])/(4*n^2 - 4*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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.