Real-rooted atlas

A038336 → x ↦ x+1 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(32 + 10\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1024 + 640\,x + 100\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
010246401001764
132768307209600100074088
210485761310720614400128000100003111696
3335544325242880032768000102400001600000100000130691232
41073741824201326592015728640006553600001536000001920000010000005489031744
53435973836875161927680704643072003670016000011468800000215040000022400000010000000230539333248
610995116277762748779069440300647710720018790481920007340032000001835008000002867200000025600000001000000009682651996416
735184372088832989560464998401236950581248009019431321600042278584320000132120576000002752512000000368640000000288000000001000000000406671383849472
8112589990684262435184372088832004947802324992000412316860416000022548578304000008455716864000002202009600000003932160000000046080000000003200000000001000000000017080198121677824
93602879701896396812384898975268864019351404648857600018141941858304000011338713661440000049606872268800000155021475840000003460300800000000540672000000000563200000000003520000000000100000000000717368321110468608
1011529215046068469764323455642275676160743093938516131840077405618595430400005442582557491200000272129127874560000099213744537600000026575110144000000051904512000000000720896000000000067584000000000038400000000000100000000000030129469486639681536
113689348814741910323214987979559889010688028102461674791895040032200737335699046400025156826043514880000014150714649477120000058961311039488000000184254096998400000004318455398400000000749731840000000000937164800000000007987200000000000416000000000000100000000000001265437718438866624512
12118059162071741130342451650883406386744524801049158569192230748160013114482114902884352000112702580674946662400007043911292184166400000330183341821132800000011792262207897600000003224446697472000000006717597286400000000010496245760000000000119275520000000000093184000000000000448000000000000010000000000000053148384174432398229504
13377789318629571617095681770887431076116955136003873816255479005839360005245792845961153740800004917930793088581632000003381077420248399872000001760977823046041600000007075357324738560000000022110491639808000000000537407782912000000000010076395929600000000001431306240000000000001490944000000000000010752000000000000004800000000000000010000000000000002232232135326160725639168
14120892581961462917470617660446290980731458735308801416709944860893564108800020660353362554697809920000209831713838446149632000001573737853788346122240000090162064539957329920000004025092166962380800000000141507146494771200000000039307540692992000000000085985245265920000000000146565758976000000000001908408320000000000000183500800000000000000122880000000000000005120000000000000001000000000000000093753749683698750476845056
Technical details
Node id
n-1b17d2b3f9a4e6b8 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-833b492313a93e6e

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:              1024,                640,                100;
  1:             32768,              30720,               9600,               1000;
  2:           1048576,            1310720,             614400,             128000,              10000;
  3:          33554432,           52428800,           32768000,           10240000,            1600000,            100000;
  4:        1073741824,         2013265920,         1572864000,          655360000,          153600000,          19200000,           1000000;
  5:       34359738368,        75161927680,        70464307200,        36700160000,        11468800000,        2150400000,         224000000,         10000000;
  6:     1099511627776,      2748779069440,      3006477107200,      1879048192000,       734003200000,      183500800000,       28672000000,       2560000000,       100000000;
  7:    35184372088832,     98956046499840,    123695058124800,     90194313216000,     42278584320000,    13212057600000,     2752512000000,     368640000000,     28800000000,     1000000000;
  8:  1125899906842624,   3518437208883200,   4947802324992000,   4123168604160000,   2254857830400000,   845571686400000,   220200960000000,   39321600000000,   4608000000000,   320000000000,   10000000000;
  9: 36028797018963968, 123848989752688640, 193514046488576000, 181419418583040000, 113387136614400000, 49606872268800000, 15502147584000000, 3460300800000000, 540672000000000, 56320000000000, 3520000000000, 100000000000;
%F Formula
The row polynomials satisfy P(n,x) = (32+10*x)*P(n-1,x) for n >= 1, with P(0,x) = 1024 + 640*x + 100*x^2.
%t Mathematica
P[0, x_] := 1024 + 640*x + 100*x^2;
P[n_, x_] := P[n, x] = Expand[(32 + 10*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1024 + 640*x + 100*x^2; for(n = 1, N, my(i = n + 1); v[i] = (32 + 10*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 - 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.