A086915 → x ↦ x+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel
Recurrence
\(\left(2\right)\,P_{n}(x) = \left(8 + 4\,x\right)\,P_{n-1}(x) + \left(6 + 4\,x\right)\,P_{n-1}'(x) + \left(1 + x\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 2\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 2 | 2 | |||||||
| 1 | 8 | 4 | 12 | ||||||
| 2 | 44 | 40 | 8 | 92 | |||||
| 3 | 304 | 384 | 144 | 16 | 848 | ||||
| 4 | 2512 | 3968 | 2112 | 448 | 32 | 9072 | |||
| 5 | 24064 | 44960 | 30400 | 9280 | 1280 | 64 | 110048 | ||
| 6 | 261536 | 558528 | 452160 | 177280 | 35520 | 3456 | 128 | 1488608 | |
| 7 | 3173888 | 7571200 | 7074816 | 3351040 | 869120 | 123648 | 8960 | 256 | 22172928 |
Compute more
?
Technical details
- Node id
n-227b620650ca6bf3· revision 0 · JSON- Definition source
derived:shift_x_plus_one- Fingerprint
f6-7b312ef315523524
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: 2; 1: 8, 4; 2: 44, 40, 8; 3: 304, 384, 144, 16; 4: 2512, 3968, 2112, 448, 32; 5: 24064, 44960, 30400, 9280, 1280, 64; 6: 261536, 558528, 452160, 177280, 35520, 3456, 128; 7: 3173888, 7571200, 7074816, 3351040, 869120, 123648, 8960, 256; 8: 42483968, 111351808, 117168128, 64540672, 20357120, 3770368, 401408, 22528, 512; 9: 621159424, 1766877696, 2056679424, 1283874816, 473647104, 107218944, 14923776, 1234944, 55296, 1024;
%F FormulaThe row polynomials satisfy 2*P(n,x) = (8+4*x)*P(n-1,x) + (6+4*x)*d/dx P(n-1,x) + (1+x)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 2.
%t MathematicaP[0, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((8 + 4*x)*P[n - 1, x] + (6 + 4*x)*D[P[n - 1, x], x] + (1 + x)*D[P[n - 1, x], {x, 2}])/(2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 2; for(n = 1, N, my(i = n + 1); v[i] = ((8 + 4*x)*v[i - 1] + (6 + 4*x)*deriv(v[i - 1], x) + (1 + x)*deriv(deriv(v[i - 1], x), x))/(2)); 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 (−∞, -1]
- Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
- Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
Proof chains ? JSON
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangle read by rows: T(n,k) = 2^k * (n!/k!)*binomial(n-1,k-1).?
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Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
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from
Triangle read by rows: T(n,k) = 2^k * (n!/k!)*binomial(n-1,k-1).?
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.