Real-rooted atlas

basis-map coefficients of A112227 → basis A013611 → x ↦ x+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(5 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 25 + 40\,x + 16\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
125401681
212530024064729
36252000240012802566561
431251250020000160006400102459049
5156257500015000016000096000307204096531441
678125437500105000014000001120000537600143360163844782969
7390625250000070000001120000011200000716800028672006553606553643046721
819531251406250045000000840000001008000008064000043008000147456002949120262144387420489
9976562578125000281250000600000000840000000806400000537600000245760000737280001310720010485763486784401
1048828125429687500171875000041250000006600000000739200000059136000003379200000135168000036044800057671680419430431381059609
112441406252343750000103125000002750000000049500000000633600000005913600000040550400000202752000007208960000173015040025165824016777216282429536481
12122070312512695312500609375000001787500000003575000000005148000000005491200000004392960000002635776000001171456000003748659200081788928001090519040671088642541865828329
13610351562568359375000355468750000113750000000025025000000004004000000000480480000000043929600000003075072000000164003840000065601536000019084083200038168166400469762048026843545622876792454961
14305175781253662109375002050781250000710937500000017062500000000300300000000004004000000000041184000000000329472000000002050048000000098402304000003578265600000954204160000176160768000201326592001073741824205891132094649
1515258789062519531250000001171875000000043750000000000113750000000000218400000000000320320000000000366080000000000329472000000000234291200000000131203072000000572522496000001908408320000046976204800008053063680008589934592042949672961853020188851841
Technical details
Node id
n-03ac2a4394de29ac · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-6b726760bd05165a

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,        40,         16;
   2:      125,       300,        240,         64;
   3:      625,      2000,       2400,       1280,        256;
   4:     3125,     12500,      20000,      16000,       6400,       1024;
   5:    15625,     75000,     150000,     160000,      96000,      30720,       4096;
   6:    78125,    437500,    1050000,    1400000,    1120000,     537600,     143360,      16384;
   7:   390625,   2500000,    7000000,   11200000,   11200000,    7168000,    2867200,     655360,      65536;
   8:  1953125,  14062500,   45000000,   84000000,  100800000,   80640000,   43008000,   14745600,    2949120,    262144;
   9:  9765625,  78125000,  281250000,  600000000,  840000000,  806400000,  537600000,  245760000,   73728000,  13107200,  1048576;
  10: 48828125, 429687500, 1718750000, 4125000000, 6600000000, 7392000000, 5913600000, 3379200000, 1351680000, 360448000, 57671680, 4194304;
%F Formula
The row polynomials satisfy P(n,x) = (5+4*x)*P(n-1,x) for n >= 2, with P(1,x) = 25 + 40*x + 16*x^2.
%t Mathematica
P[1, x_] := 25 + 40*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[(5 + 4*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 + 40*x + 16*x^2; for(n = 2, N, my(i = n); v[i] = (5 + 4*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.