A008955 → basis A013609 → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(1 - 2\,n + n^{2}\right)\,P_{n}(x) = \left(-4\,n^{2} - 16\,n^{2} x + 36\,n^{2} x^{2} + 8\,n^{3} + 48\,n^{3} x - 144\,n^{3} x^{2} - 8\,n^{4} - 56\,n^{4} x + 216\,n^{4} x^{2} + 4\,n^{5} + 32\,n^{5} x - 144\,n^{5} x^{2} - n^{6} - 8\,n^{6} x + 36\,n^{6} x^{2}\right)\,P_{n-2}(x) + \left(16\,n^{2} x^{2} + 16\,n^{2} x^{3} - 48\,n^{3} x^{2} - 64\,n^{3} x^{3} + 56\,n^{4} x^{2} + 96\,n^{4} x^{3} - 32\,n^{5} x^{2} - 64\,n^{5} x^{3} + 8\,n^{6} x^{2} + 16\,n^{6} x^{3}\right)\,P_{n-2}'(x) + \left(1 - 2\,n + 4\,n^{2} + 8\,n^{2} x - 4\,n^{3} - 16\,n^{3} x + 2\,n^{4} + 8\,n^{4} x\right)\,P_{n-1}(x) + \left(-8\,n^{2} x^{2} + 16\,n^{3} x^{2} - 8\,n^{4} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 8\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | ||||||||||||||
| 1 | 2 | 8 | 10 | |||||||||||||
| 2 | 10 | 104 | 144 | 258 | ||||||||||||
| 3 | 100 | 1760 | 5652 | 4608 | 12120 | |||||||||||
| 4 | 1700 | 42720 | 222804 | 399872 | 230400 | 897496 | ||||||||||
| 5 | 44200 | 1450720 | 10598904 | 30201472 | 37230400 | 16588800 | 96114496 | |||||||||
| 6 | 1635400 | 66406240 | 627176088 | 2474114176 | 4775190400 | 4473833472 | 1625702400 | 14044058176 | ||||||||
| 7 | 81770000 | 3961388800 | 46001380320 | 232973720576 | 617608253200 | 897566542848 | 678044795904 | 208089907200 | 2684327758848 | |||||||
| 8 | 5315050000 | 299356512000 | 4130969695200 | 25611161492480 | 86739280573200 | 172179378514944 | 200448949421056 | 126883863724032 | 33710564966400 | 650008839949312 | ||||||
| 9 | 435834100000 | 27991386384000 | 447854963630400 | 3289834514600960 | 13595446259786400 | 34353248410341504 | 54402366815071744 | 52817838204911616 | 28779423511412736 | 6742112993280000 | 194456552893519360 | |||||
| 10 | 44019244100000 | 3175797304784000 | 57829475199470400 | 491510606376616960 | 2401213358051226400 | 7385166612262975104 | 14846356671137433344 | 19545832173653590016 | 16276237070270939136 | 7786983908966400000 | 1631591344373760000 | 70425941036841295360 | ||||
| 11 | 5370347780200000 | 430057899472448000 | 8784417606790276800 | 84843819308208312320 | 478800477718407908800 | 1737765157709774338560 | 4243847338551387945280 | 7077252180748525076480 | 7972222833009893191680 | 5812789038135341875200 | 2479238772237139968000 | 469898307179642880000 | 30365877770452364421120 | |||
| 12 | 778700428129000000 | 68545036066295360000 | 1552418071842736440000 | 16799975614366827008000 | 107605787957862887320000 | 450544081987295407028736 | 1296561805912182792781120 | 2622581038538723536240640 | 3735630730669271903170560 | 3677422528933142151168000 | 2385130345983789232029696 | 917878976542855397376000 | 158825627826719293440000 | 15370602641774546588362752 | ||
| 13 | 132379072781930000000 | 12705459110100619200000 | 316039572141682816080000 | 3788826624722973776640000 | 27165471074174171358000000 | 128966383052691243711275520 | 427690814946004033973066112 | 1018231123194283780210728960 | 1756948219106169551276436480 | 2183980579259638984552448000 | 1909464424700320646429736960 | 1115453179729004339052675072 | 391105099078326904946688000 | 62259646108073963028480000 | 9025382644283735157681455104 | |
| 14 | 26078677338040210000000 | 2710545830811888222400000 | 73466010647020260902160000 | 966643311345163014270720000 | 7672254109255133195718000000 | 40740742573330011259285117440 | 153065820702032056059532362624 | 419569228521627970095723454464 | 851288715158679442058505891840 | 1280520349286517354154811392000 | 1412070160020395130499575316480 | 1110533101575900630410449649664 | 590218136814868452035342106624 | 190072173788533249774977024000 | 28016840748633283362816000000 | 6084810369246273046110341595136 |
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Technical details
- Node id
n-50073e7cd4eca286· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-cf0961e8793f3f0e
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): no
Matches ?
No OEIS matches.
Submission text ?
Atlas rows n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 1; 1: 2, 8; 2: 10, 104, 144; 3: 100, 1760, 5652, 4608; 4: 1700, 42720, 222804, 399872, 230400; 5: 44200, 1450720, 10598904, 30201472, 37230400, 16588800; 6: 1635400, 66406240, 627176088, 2474114176, 4775190400, 4473833472, 1625702400; 7: 81770000, 3961388800, 46001380320, 232973720576, 617608253200, 897566542848, 678044795904, 208089907200; 8: 5315050000, 299356512000, 4130969695200, 25611161492480, 86739280573200, 172179378514944, 200448949421056, 126883863724032, 33710564966400; 9: 435834100000, 27991386384000, 447854963630400, 3289834514600960, 13595446259786400, 34353248410341504, 54402366815071744, 52817838204911616, 28779423511412736, 6742112993280000;
%F FormulaThe row polynomials satisfy (n^2-2*n+1)*P(n,x) = -(n^6-4*n^5+8*n^4-8*n^3+4*n^2+(8*n^6-32*n^5+56*n^4-48*n^3+16*n^2)*x-(36*n^6-144*n^5+216*n^4-144*n^3+36*n^2)*x^2)*P(n-2,x) + ((8*n^6-32*n^5+56*n^4-48*n^3+16*n^2)*x^2+(16*n^6-64*n^5+96*n^4-64*n^3+16*n^2)*x^3)*d/dx P(n-2,x) + (2*n^4-4*n^3+4*n^2-2*n+1+(8*n^4-16*n^3+8*n^2)*x)*P(n-1,x) - (8*n^4-16*n^3+8*n^2)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 8*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 2 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^6 - 4*n^5 + 8*n^4 - 8*n^3 + 4*n^2 + (8*n^6 - 32*n^5 + 56*n^4 - 48*n^3 + 16*n^2)*x - (36*n^6 - 144*n^5 + 216*n^4 - 144*n^3 + 36*n^2)*x^2)*P[n - 2, x] + ((8*n^6 - 32*n^5 + 56*n^4 - 48*n^3 + 16*n^2)*x^2 + (16*n^6 - 64*n^5 + 96*n^4 - 64*n^3 + 16*n^2)*x^3)*D[P[n - 2, x], x] + (2*n^4 - 4*n^3 + 4*n^2 - 2*n + 1 + (8*n^4 - 16*n^3 + 8*n^2)*x)*P[n - 1, x] - (8*n^4 - 16*n^3 + 8*n^2)*x^2*D[P[n - 1, x], x])/(n^2 - 2*n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^6 - 4*n^5 + 8*n^4 - 8*n^3 + 4*n^2 + (8*n^6 - 32*n^5 + 56*n^4 - 48*n^3 + 16*n^2)*x - (36*n^6 - 144*n^5 + 216*n^4 - 144*n^3 + 36*n^2)*x^2)*v[i - 2] + ((8*n^6 - 32*n^5 + 56*n^4 - 48*n^3 + 16*n^2)*x^2 + (16*n^6 - 64*n^5 + 96*n^4 - 64*n^3 + 16*n^2)*x^3)*deriv(v[i - 2], x) + (2*n^4 - 4*n^3 + 4*n^2 - 2*n + 1 + (8*n^4 - 16*n^3 + 8*n^2)*x)*v[i - 1] - (8*n^4 - 16*n^3 + 8*n^2)*x^2*deriv(v[i - 1], x))/(n^2 - 2*n + 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 have one sign
derived from Laguerre’s theorem on multiplier sequences
-
Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) 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
- [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
- [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.