Real-rooted atlas

basis-map coefficients of A112227 → basis A013611 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(5 + 12\,x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 25 + 120\,x + 144\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
125120144289
2125900216017284913
3625600021600345602073683521
43125375001800004320005184002488321419857
5156252250001350000432000077760007464960298598424137569
67812513125009450000378000009072000013063680010450944035831808410338673
73906257500000630000003024000009072000001741824000209018880014332723204299816966975757441
819531254218750040500000022680000008164800000195955200003135283200032248627200193491763205159780352118587876497

All 15 rows

Technical details
Node id
n-5d4a920bb02efecc · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-f055f948fcf55643

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 1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:       25,        120,         144;
   2:      125,        900,        2160,         1728;
   3:      625,       6000,       21600,        34560,        20736;
   4:     3125,      37500,      180000,       432000,       518400,        248832;
   5:    15625,     225000,     1350000,      4320000,      7776000,       7464960,       2985984;
   6:    78125,    1312500,     9450000,     37800000,     90720000,     130636800,     104509440,      35831808;
   7:   390625,    7500000,    63000000,    302400000,    907200000,    1741824000,    2090188800,    1433272320,     429981696;
   8:  1953125,   42187500,   405000000,   2268000000,   8164800000,   19595520000,   31352832000,   32248627200,   19349176320,    5159780352;
   9:  9765625,  234375000,  2531250000,  16200000000,  68040000000,  195955200000,  391910400000,  537477120000,  483729408000,  257989017600,   61917364224;
  10: 48828125, 1289062500, 15468750000, 111375000000, 534600000000, 1796256000000, 4311014400000, 7390310400000, 8868372480000, 7094697984000, 3405455032320, 743008370688;
%F Formula
The row polynomials satisfy P(n,x) = (5+12*x)*P(n-1,x) for n >= 2, with P(1,x) = 25 + 120*x + 144*x^2.
%t Mathematica
P[1, x_] := 25 + 120*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[(5 + 12*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 25 + 120*x + 144*x^2; for(n = 2, N, my(i = n); v[i] = (5 + 12*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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.