Real-rooted atlas

A013622 → basis A013611 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(28 + 40\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1284068
2784224016004624
3219529408013440064000314432
4614656351232075264007168000256000021381376
5172103681229312003512320005017600003584000001024000001453933568
6481890304413048832014751744000280985600003010560000017203200000409600000098867482624
7134929285121349292851205782683648001376829440000196689920000016859136000008028160000001638400000006722988818432
83778019983364317737123840215886856192006168195891200011014635520000012588154880000089915392000000367001600000006553600000000457163239653376
91057845595340813600871940096077719268229120025906422743040005551376302080000793053757440000075528929280000004624220160000000165150720000000026214400000000031087100296429568
1029619676669542442313823813632002720174388019200010362569097216000025906422743040000044411010416640000052870250496000000043159388160000000023121100800000000073400320000000000104857600000000002113922820157210624
11829350946747187213032657734598656093090412389990400039895891024281600001139882600693760000022797652013875200000325680743055360000003323272888320000000023737663488000000000113036492800000000003229614080000000000419430400000000000143746751770690322432
1223221826508921241639808845443864985603127837856303677440014894465982398464000047875069229137920000010942872966660096000001823812161110016000000223323938095104000000019939637329920000000001266008719360000000000542575165440000000000140928614400000000000167772160000000000009774779120406941925376
1365021114224979476481207534978463904563201035029981540489625600542158561759304089600019362805777118003200000497900719983034368000009483823237772083200000013548318911102976000000014516055976181760000000011520679346176000000000065832453406720000000000256490078208000000000006106906624000000000000671088640000000000000664684980187672050925568
14182059119829942534144364118239659885068288033810979396989327769600193205596554224730112000759021986463025725440000216863424703721635840000046470733865083207680000007587058590217666560000000948382323777208320000000090322126074019840000000006451580433858560000000000335147035525120000000000011969536983040000000000002630667468800000000000002684354560000000000000045198578652761699462938624
Technical details
Node id
n-8a39251d62965edd · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-0fbe6fa5f58efdcf

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:             28,              40;
  2:            784,            2240,            1600;
  3:          21952,           94080,          134400,            64000;
  4:         614656,         3512320,         7526400,          7168000,          2560000;
  5:       17210368,       122931200,       351232000,        501760000,        358400000,        102400000;
  6:      481890304,      4130488320,     14751744000,      28098560000,      30105600000,      17203200000,       4096000000;
  7:    13492928512,    134929285120,    578268364800,    1376829440000,    1966899200000,    1685913600000,     802816000000,     163840000000;
  8:   377801998336,   4317737123840,  21588685619200,   61681958912000,  110146355200000,  125881548800000,   89915392000000,   36700160000000,    6553600000000;
  9: 10578455953408, 136008719400960, 777192682291200, 2590642274304000, 5551376302080000, 7930537574400000, 7552892928000000, 4624220160000000, 1651507200000000, 262144000000000;
%F Formula
The row polynomials satisfy P(n,x) = (28+40*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[(28 + 40*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] = (28 + 40*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.