Real-rooted atlas

A038208 → basis A013610 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1101828
2100360324784
3100054009720583221952
41000072000194400233280104976614656
5100000900000324000058320005248800188956817210368
61000000108000004860000011664000015746400011337408034012224481890304
710000000126000000680400000204120000036741600003968092800238085568061222003213492928512
8100000000144000000090720000003265920000073483200000105815808000952342272004897760256011019960576377801998336
9100000000016200000000116640000000489888000000132269760000023808556800002857026816000220399211520099179645184019835929036810578455953408
1010000000000180000000000145800000000069984000000002204496000000047617113600000714256704000007346640384000049589822592000198359290368003570467226624296196766695424
11100000000000198000000000017820000000000962280000000003464208000000008729804160000001571364748800000202032610560000018182934950400001090976097024000392751394928640642684100792328293509467471872
121000000000000216000000000002138400000000001283040000000000519631200000000014965378560000000314272949760000004848782653440000054548804851200000436390438809600002356508369571840077122092095078401156831381426176232218265089212416
1310000000000000234000000000000252720000000000016679520000000000750578400000000002431874016000000005836497638400000001050569574912000000141826892613120000014182689261312000001021153626814464000501293598618009600150388079585402880208229648656711686502111422497947648
1410000000000000025200000000000002948400000000000021228480000000000010508097600000000003782915136000000000102138708672000000002101139149824000000033092941609728000000397115299316736000003574037693850624000023393701268840448000105271655709782016002915215081193963520374813367582081024182059119829942534144
Technical details
Node id
n-1273b38f23038cfd · revision 0 · JSON
Definition source
derived:basis_map_A013610
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:         10,          18;
  2:        100,         360,          324;
  3:       1000,        5400,         9720,         5832;
  4:      10000,       72000,       194400,       233280,        104976;
  5:     100000,      900000,      3240000,      5832000,       5248800,       1889568;
  6:    1000000,    10800000,     48600000,    116640000,     157464000,     113374080,      34012224;
  7:   10000000,   126000000,    680400000,   2041200000,    3674160000,    3968092800,    2380855680,     612220032;
  8:  100000000,  1440000000,   9072000000,  32659200000,   73483200000,  105815808000,   95234227200,   48977602560,  11019960576;
  9: 1000000000, 16200000000, 116640000000, 489888000000, 1322697600000, 2380855680000, 2857026816000, 2203992115200, 991796451840, 198359290368;
%F Formula
The row polynomials satisfy P(n,x) = (10+18*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 + 18*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 + 18*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.