Real-rooted atlas

A130123 → 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

\(P_{n}(x) = \left(10 + 8\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
110818
210016064324
31000240019205125832
4100003200038400204804096104976
5100000400000640000512000204800327681889568
6100000048000009600000102400006144000196608026214434012224
7100000005600000013440000017920000014336000068812800183500802097152612220032
810000000064000000017920000002867200000286720000018350080007340032001677721601677721611019960576
910000000007200000000230400000004300800000051609600000412876800002202009600075497472001509949440134217728198359290368
101000000000080000000000288000000000614400000000860160000000825753600000550502400000251658240000754974720001342177280010737418243570467226624
111000000000008800000000003520000000000844800000000013516800000000151388160000001211105280000069206016000002768240640000738197504000118111600640858993459264268410079232
121000000000000960000000000042240000000000112640000000000202752000000000259522560000000242221056000000166094438400000830472192000002952790016000070866960384001030792151040687194767361156831381426176
131000000000000010400000000000049920000000000014643200000000002928640000000000421724160000000044983910400000003598712832000000215922769920000095965675520000030709016166400067001489817600893353197568054975581388820822964865671168
141000000000000001120000000000000582400000000000018636800000000000410009600000000006560153600000000078721843200000000719742566400000005038197964800000026870389145600000107481556582400003126736191488000625347238297600769658139443204398046511104374813367582081024
Technical details
Node id
n-881b87f0bbbf8ba6 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-6b726760bd05165a

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 0)
  • 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:          1;
  1:         10,          8;
  2:        100,        160,          64;
  3:       1000,       2400,        1920,         512;
  4:      10000,      32000,       38400,       20480,        4096;
  5:     100000,     400000,      640000,      512000,      204800,       32768;
  6:    1000000,    4800000,     9600000,    10240000,     6144000,     1966080,      262144;
  7:   10000000,   56000000,   134400000,   179200000,   143360000,    68812800,    18350080,    2097152;
  8:  100000000,  640000000,  1792000000,  2867200000,  2867200000,  1835008000,   734003200,  167772160,   16777216;
  9: 1000000000, 7200000000, 23040000000, 43008000000, 51609600000, 41287680000, 22020096000, 7549747200, 1509949440, 134217728;
%F Formula
The row polynomials satisfy P(n,x) = (10+8*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(10 + 8*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (10 + 8*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. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [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
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.