Real-rooted atlas

A038319 → basis A013609 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(10 + 16\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 100 + 320\,x + 256\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0100320256676
1100048007680409617576
2100006400015360016384065536456976
3100000800000256000040960003276800104857611881376
4100000096000003840000081920000983040006291456016777216308915776
51000000011200000053760000014336000002293760000220200960011744051202684354568031810176
61000000001280000000716800000022937600000458752000005872025600046976204800214748364804294967296208827064576
71000000000144000000009216000000034406400000082575360000013212057600001409286144000966367641600386547056640687194767365429503678976
81000000000016000000000011520000000004915200000000137625600000002642411520000035232153600000322122547200001932735283200068719476736001099511627776141167095653376
910000000000017600000000001408000000000067584000000000216268800000000484442112000000775107379200000885837004800000708669603840000377957122048000120946279055360175921860444163670344486987776
1010000000000001920000000000016896000000000090112000000000032440320000000008304721920000000155021475840000002126008811520000021260088115200000151182848819200007256776743321600211106232532992028147497671065695428956661682176
1110000000000000208000000000000199680000000000011714560000000000468582400000000001349517312000000002878970265600000004606352424960000005527622909952000004913442586624000003144603255439360001372190511464448003659174697238528045035996273704962481152873203736576
12100000000000000224000000000000023296000000000000149094400000000000656015360000000000209924915200000000050381979648000000009212704849920000000128977867898880000001375763924254720000011006111394037760000640355572016742400025614222880669696006305039478318694407205759403792793664509974703297150976
13100000000000000024000000000000000268800000000000000186368000000000000089456640000000000003148873728000000000083969966080000000000172738215936000000000276381145497600000000343940981063680000000330183341821132800000240133339506278400000128071114403348480000472877960873902080001080863910568919040011529215046068469761677259342285725925376
1410000000000000000256000000000000000307200000000000000022937600000000000000119275520000000000000458017996800000000000134351945728000000000030709016166400000000005527622909952000000000786150813859840000000088048891152302080000007684266864200908800000512284457613393920000025220157913274777600008646911284551352320001844674407370955161601844674407370955161643608742899428874059776
Technical details
Node id
n-b8de3d8ed94eb089 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-4b28493176e02e0b

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:          100,           320,            256;
  1:         1000,          4800,           7680,           4096;
  2:        10000,         64000,         153600,         163840,           65536;
  3:       100000,        800000,        2560000,        4096000,         3276800,         1048576;
  4:      1000000,       9600000,       38400000,       81920000,        98304000,        62914560,        16777216;
  5:     10000000,     112000000,      537600000,     1433600000,      2293760000,      2202009600,      1174405120,       268435456;
  6:    100000000,    1280000000,     7168000000,    22937600000,     45875200000,     58720256000,     46976204800,     21474836480,      4294967296;
  7:   1000000000,   14400000000,    92160000000,   344064000000,    825753600000,   1321205760000,   1409286144000,    966367641600,    386547056640,     68719476736;
  8:  10000000000,  160000000000,  1152000000000,  4915200000000,  13762560000000,  26424115200000,  35232153600000,  32212254720000,  19327352832000,   6871947673600,   1099511627776;
  9: 100000000000, 1760000000000, 14080000000000, 67584000000000, 216268800000000, 484442112000000, 775107379200000, 885837004800000, 708669603840000, 377957122048000, 120946279055360, 17592186044416;
%F Formula
The row polynomials satisfy P(n,x) = (10+16*x)*P(n-1,x) for n >= 1, with P(0,x) = 100 + 320*x + 256*x^2.
%t Mathematica
P[0, x_] := 100 + 320*x + 256*x^2;
P[n_, x_] := P[n, x] = Expand[(10 + 16*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] = 100 + 320*x + 256*x^2; for(n = 1, N, my(i = n + 1); v[i] = (10 + 16*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.