Real-rooted atlas

A038255 → basis A013610 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(10 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
110616
210012036256
31000180010802164096
41000024000216008640129665536
51000003000003600002160006480077761048576
6100000036000005400000432000019440004665604665616777216
71000000042000000756000007560000045360000163296003265920279936268435456
8100000000480000000100800000012096000009072000004354560001306368002239488016796164294967296
9100000000054000000001296000000018144000000163296000009797760000391910400010077696001511654401007769668719476736
101000000000060000000000162000000000259200000000272160000000195955200000979776000003359232000075582720001007769600604661761099511627776
111000000000006600000000001980000000000356400000000042768000000003592512000000215550720000092378880000027713664000055427328000665127936036279705617592186044416
121000000000000720000000000023760000000000475200000000006415200000000061585920000000431101440000002217093120000083140992000002217093120000399076761600435356467202176782336281474976710656
131000000000000078000000000000280800000000000617760000000000926640000000000100077120000000080061696000000048037017600000021616657920000072055526400000172933263360002829817036800282981703680130606940164503599627370496
14100000000000000840000000000000327600000000000078624000000000001297296000000000015567552000000000140107968000000009607403520000000504388684800000020175547392000006052664217600001320581283840001980871925760018284971622407836416409672057594037927936
Technical details
Node id
n-e7973e04614ef858 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-b1029f17d1b64336

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,          6;
  2:        100,        120,          36;
  3:       1000,       1800,        1080,         216;
  4:      10000,      24000,       21600,        8640,        1296;
  5:     100000,     300000,      360000,      216000,       64800,       7776;
  6:    1000000,    3600000,     5400000,     4320000,     1944000,     466560,      46656;
  7:   10000000,   42000000,    75600000,    75600000,    45360000,   16329600,    3265920,     279936;
  8:  100000000,  480000000,  1008000000,  1209600000,   907200000,  435456000,  130636800,   22394880,   1679616;
  9: 1000000000, 5400000000, 12960000000, 18144000000, 16329600000, 9797760000, 3919104000, 1007769600, 151165440, 10077696;
%F Formula
The row polynomials satisfy P(n,x) = (10+6*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 + 6*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 + 6*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.