Triangle read by rows: T(n, m) = S1(n, m)*6^(n-m), where S1 are the signed Stirling numbers of first kind A008275 (n >= 1, 1 <= m <= n).
A051151 (tail) ↗ only rational roots
Proved for all n all roots real, in [0, ∞)alternating coefficient signsinterlacing (weak)
Recurrence from A051151
\(P_{n}(x) = \left(-12 + x - 6\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 72 - 18\,x + x^{2}\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | x15 | x16 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 72 | -18 | 1 | 55 | ||||||||||||||
| 1 | -1296 | 396 | -36 | 1 | -935 | |||||||||||||
| 2 | 31104 | -10800 | 1260 | -60 | 1 | 21505 | ||||||||||||
| 3 | -933120 | 355104 | -48600 | 3060 | -90 | 1 | -623645 | |||||||||||
| 4 | 33592320 | -13716864 | 2104704 | -158760 | 6300 | -126 | 1 | 21827575 | ||||||||||
| 5 | -1410877440 | 609700608 | -102114432 | 8772624 | -423360 | 11592 | -168 | 1 | -894930575 | |||||||||
| 6 | 67722117120 | -30676506624 | 5511193344 | -523200384 | 29093904 | -979776 | 19656 | -216 | 1 | 42061737025 | ||||||||
| 7 | -3656994324480 | 1724253474816 | -328280947200 | 33764014080 | -2094271200 | 82001808 | -2041200 | 31320 | -270 | 1 | -2229272062325 | |||||||
| 8 | 219419659468800 | -107112202813440 | 21421110306816 | -2354121792000 | 159420286080 | -7014379680 | 204473808 | -3920400 | 47520 | -330 | 1 | 131527051677175 | ||||||
| 9 | -14481697524940800 | 7288825045155840 | -1520905483063296 | 176793148578816 | -12875860673280 | 622369344960 | -20509651008 | 463220208 | -7056720 | 69300 | -396 | 1 | -8549258359016375 | |||||
| 10 | 1042682221795737600 | -539277100776161280 | 116794019825713152 | -14250012180738048 | 1103855117054976 | -57686453510400 | 2099064217536 | -53861505984 | 971304048 | -12046320 | 97812 | -468 | 1 | 606997343490162625 | ||||
| 11 | -81329213300067532800 | 43106296082336317440 | -9649210647181787136 | 1228294969923280896 | -100350711311026176 | 5603398490866176 | -221413462478208 | 6300261684288 | -129623221728 | 1910917008 | -19675656 | 134316 | -546 | 1 | -46738795448742522125 | |||
| 12 | 6831653917205672755200 | -3702258084216318197760 | 853639990445606436864 | -112825988120737382400 | 9657754720049479680 | -571036184543784960 | 24202129339035648 | -750635443958400 | 17188612309440 | -290140250400 | 3563672112 | -30958200 | 180180 | -630 | 1 | 3879320022245629336375 | ||
| 13 | -614848852548510547968000 | 340034881496674310553600 | -80529857224320897515520 | 11007978921311970852864 | -982023912925190553600 | 61051011328990126080 | -2749227825056993280 | 91759319295291648 | -2297610551808000 | 43301234845440 | -610870740480 | 6349910112 | -47174400 | 236880 | -720 | 1 | -345259481979861010937375 | |
| 14 | 59025489844657012604928000 | -33258197476229244361113600 | 8070901175031480472043520 | -1137295833670270099390464 | 105282274562130263998464 | -6842921000508242657280 | 324976882534461480960 | -11558122477404991488 | 312329932268859648 | -6454529096970240 | 101944825931520 | -1220462111232 | 10878652512 | -69914880 | 306000 | -816 | 1 | 32799650788086796039050625 |
Compute more
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Technical details
- Node id
n-380b79aacd7d0dd2· revision 0 · JSON- Definition source
oeis_catalog:A051151- Fingerprint
f6-6b6154934dc9ab46
Observed on the stored rows ?
- Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
- Palindromic rows: no
- γ-positive (gamma_positive): no
- Nonnegative coefficients (
nonn): no - 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 ExampleTriangle begins: 0: 72, -18, 1; 1: -1296, 396, -36, 1; 2: 31104, -10800, 1260, -60, 1; 3: -933120, 355104, -48600, 3060, -90, 1; 4: 33592320, -13716864, 2104704, -158760, 6300, -126, 1; 5: -1410877440, 609700608, -102114432, 8772624, -423360, 11592, -168, 1; 6: 67722117120, -30676506624, 5511193344, -523200384, 29093904, -979776, 19656, -216, 1; 7: -3656994324480, 1724253474816, -328280947200, 33764014080, -2094271200, 82001808, -2041200, 31320, -270, 1; 8: 219419659468800, -107112202813440, 21421110306816, -2354121792000, 159420286080, -7014379680, 204473808, -3920400, 47520, -330, 1; 9: -14481697524940800, 7288825045155840, -1520905483063296, 176793148578816, -12875860673280, 622369344960, -20509651008, 463220208, -7056720, 69300, -396, 1;
%F FormulaThe row polynomials satisfy P(n,x) = -(6*n+12-x)*P(n-1,x) for n >= 1, with P(0,x) = 72 - 18*x + x^2.
%t MathematicaP[0, x_] := 72 - 18*x + x^2;
P[n_, x_] := P[n, x] = Expand[-(6*n + 12 - x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 72 - 18*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = -(6*n + 12 - 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
- Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in [0, +∞)
-
Coefficients of one sign?
Every nonzero row, n ≥ 0: the coefficients of P_n(-x) have one sign
derived from Iterated real-rootedness preservers
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 1
Proof chains ? JSON
References
- [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
- [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
- [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.