Real-rooted atlas

A013622 → basis A013609 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(18 + 10\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1181028
2324360100784
3583297205400100021952
41049762332801944007200010000614656
5188956852488005832000324000090000010000017210368
63401222411337408015746400011664000048600000108000001000000481890304
761222003223808556803968092800367416000020412000006804000001260000001000000013492928512
8110199605764897760256095234227200105815808000734832000003265920000090720000001440000000100000000377801998336
9198359290368991796451840220399211520028570268160002380855680000132269760000048988800000011664000000016200000000100000000010578455953408
1035704672266241983592903680049589822592000734664038400007142567040000047617113600000220449600000006998400000000145800000000018000000000010000000000296196766695424
11642684100792323927513949286401090976097024000181829349504000020203261056000001571364748800000872980416000000346420800000000962280000000001782000000000019800000000001000000000008293509467471872
121156831381426176771220920950784023565083695718400436390438809600005454880485120000048487826534400000314272949760000001496537856000000051963120000000001283040000000000213840000000000216000000000001000000000000232218265089212416
1320822964865671168150388079585402880501293598618009600102115362681446400014182689261312000001418268926131200000105056957491200000058364976384000000024318740160000000075057840000000000166795200000000002527200000000000234000000000000100000000000006502111422497947648
1437481336758208102429152150811939635201052716557097820160023393701268840448000357403769385062400003971152993167360000033092941609728000000210113914982400000001021387086720000000037829151360000000001050809760000000000212284800000000000294840000000000002520000000000000100000000000000182059119829942534144
Technical details
Node id
n-3d8880ae4f67526d · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-dca555f310728b4d

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:           18,           10;
  2:          324,          360,           100;
  3:         5832,         9720,          5400,          1000;
  4:       104976,       233280,        194400,         72000,         10000;
  5:      1889568,      5248800,       5832000,       3240000,        900000,        100000;
  6:     34012224,    113374080,     157464000,     116640000,      48600000,      10800000,      1000000;
  7:    612220032,   2380855680,    3968092800,    3674160000,    2041200000,     680400000,    126000000,     10000000;
  8:  11019960576,  48977602560,   95234227200,  105815808000,   73483200000,   32659200000,   9072000000,   1440000000,   100000000;
  9: 198359290368, 991796451840, 2203992115200, 2857026816000, 2380855680000, 1322697600000, 489888000000, 116640000000, 16200000000, 1000000000;
%F Formula
The row polynomials satisfy P(n,x) = (18+10*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[(18 + 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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (18 + 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. [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.