Real-rooted atlas

basis-map coefficients of A026729 → basis A013611 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(5 + 12\,x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
251217
325120144289
4125900216017284913
5625600021600345602073683521
63125375001800004320005184002488321419857
7156252250001350000432000077760007464960298598424137569
87812513125009450000378000009072000013063680010450944035831808410338673
93906257500000630000003024000009072000001741824000209018880014332723204299816966975757441
1019531254218750040500000022680000008164800000195955200003135283200032248627200193491763205159780352118587876497
11976562523437500025312500001620000000068040000000195955200000391910400000537477120000483729408000257989017600619173642242015993900449
124882812512890625001546875000011137500000053460000000017962560000004311014400000739031040000088683724800007094697984000340545503232074300837068834271896307633
132441406257031250000928125000007425000000004009500000000153964800000004311014400000088683724800000133025587200000141893959680000102163650969600445805022412808916100448256582622237229761
1412207031253808593750054843750000048262500000002895750000000012509640000000040030848000000096074035200000017293326336000002305776844800000221354577100800014488663228416005795465291366401069932053790729904578032905937
15610351562520507812500031992187500003071250000000020270250000000097297200000000035026992000000009607403520000000201755473920000003228087582720000038737050992640000338068808663040002028412851978240074895243765350401283918464548864168377826559400929
Technical details
Node id
n-84c9a70772b28ab4 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-f055f948fcf55643

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 -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:       1;
   2:       5,       12;
   3:      25,      120,       144;
   4:     125,      900,      2160,       1728;
   5:     625,     6000,     21600,      34560,      20736;
   6:    3125,    37500,    180000,     432000,     518400,      248832;
   7:   15625,   225000,   1350000,    4320000,    7776000,     7464960,     2985984;
   8:   78125,  1312500,   9450000,   37800000,   90720000,   130636800,   104509440,    35831808;
   9:  390625,  7500000,  63000000,  302400000,  907200000,  1741824000,  2090188800,  1433272320,   429981696;
  10: 1953125, 42187500, 405000000, 2268000000, 8164800000, 19595520000, 31352832000, 32248627200, 19349176320, 5159780352;
%F Formula
The row polynomials satisfy P(n,x) = (5+12*x)*P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
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] = 1; 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.