Real-rooted atlas

A013622 → basis A013610 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(15 + 8\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
115823
222524064529
333755400288051212167
45062510800086400307204096279841
5759375202500021600001152000307200327686436343
611390625364500004860000034560000138240002949120262144148035889
717085937563787500010206000009072000004838400001548288002752512020971523404825447
8256289062510935000000204120000002177280000014515200000619315200016515072002516582401677721678310985281
938443359375184528125000393660000000489888000000391910400000209018880000743178240001698693120022649241601342177281801152661463
10576650390625307546875000073811250000001049760000000097977600000006270566400000278691840000084934656000016986931200020132659200107374182441426511213649
118649755859375507452343750001353206250000002165130000000002309472000000001724405760000009196830720000035035545600000934281216000016609443840001771674009608589934592952809757913927
12129746337890625830376562500000243577125000000043302600000000005196312000000000443418624000000027590492160000001261279641600000420426547200000996566630400001594506608640015461882265606871947673621914624432020321
1319461950683593751349361914062500043179581250000000844400700000000001125867600000000001080832896000000007685922816000000040991588352000000163966353408000004858262323200000103642929561600015075335208960013400297963520549755813888504036361936467383
14291929260253906252179738476562500007556426718750000001612037700000000000236432196000000000025219434240000000002017554739200000000122974765056000000057388223692800000020404701757440000054412538019840000105527346462720001407031286169600115448720916480439804651110411592836324538749809
Technical details
Node id
n-dd3479747c695235 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-dd0a803b420669cd

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:          15,            8;
  2:         225,          240,           64;
  3:        3375,         5400,         2880,          512;
  4:       50625,       108000,        86400,        30720,         4096;
  5:      759375,      2025000,      2160000,      1152000,       307200,        32768;
  6:    11390625,     36450000,     48600000,     34560000,     13824000,      2949120,      262144;
  7:   170859375,    637875000,   1020600000,    907200000,    483840000,    154828800,    27525120,     2097152;
  8:  2562890625,  10935000000,  20412000000,  21772800000,  14515200000,   6193152000,  1651507200,   251658240,   16777216;
  9: 38443359375, 184528125000, 393660000000, 489888000000, 391910400000, 209018880000, 74317824000, 16986931200, 2264924160, 134217728;
%F Formula
The row polynomials satisfy P(n,x) = (15+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[(15 + 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] = (15 + 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.