A376827 → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(n - 5\,n x + 5\,n x^{2} - n x^{3} + 6\,n^{2} x - 17\,n^{2} x^{2} + 5\,n^{2} x^{3} + 11\,n^{3} x^{2} - 11\,n^{3} x^{3} + n^{3} x^{4} + 6\,n^{4} x^{3} - 2\,n^{4} x^{4} + n^{5} x^{4}\right)\,P_{n}(x) = \left(-3 - 9\,x - 6\,x^{2} + 6\,x^{3} + 9\,x^{4} + 3\,x^{5} + 3\,n - 9\,n x - 45\,n x^{2} - 57\,n x^{3} - 30\,n x^{4} - 6\,n x^{5} + 18\,n^{2} x + 18\,n^{2} x^{2} - 36\,n^{2} x^{3} - 57\,n^{2} x^{4} - 24\,n^{2} x^{5} - 3\,n^{2} x^{6} + 33\,n^{3} x^{2} + 69\,n^{3} x^{3} + 36\,n^{3} x^{4} - 3\,n^{3} x^{5} - 3\,n^{3} x^{6} + 18\,n^{4} x^{3} + 39\,n^{4} x^{4} + 24\,n^{4} x^{5} + 3\,n^{4} x^{6} + 3\,n^{5} x^{4} + 6\,n^{5} x^{5} + 3\,n^{5} x^{6}\right)\,P_{n-2}(x) + \left(-1 + x + 8\,x^{2} + 8\,x^{3} + x^{4} - x^{5} + 2\,n - 8\,n x - 16\,n x^{2} - 4\,n x^{3} + 4\,n x^{4} + 2\,n x^{5} + 12\,n^{2} x - 11\,n^{2} x^{2} - 33\,n^{2} x^{3} - 8\,n^{2} x^{4} + 2\,n^{2} x^{5} + 22\,n^{3} x^{2} + 12\,n^{3} x^{3} - 14\,n^{3} x^{4} - 4\,n^{3} x^{5} + 12\,n^{4} x^{3} + 11\,n^{4} x^{4} - n^{4} x^{5} + 2\,n^{5} x^{4} + 2\,n^{5} x^{5}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 4\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 1 | 4 | 5 | ||||||
| 2 | 3 | 24 | 27 | 54 | |||||
| 3 | 7 | 84 | 189 | 112 | 392 | ||||
| 4 | 19 | 304 | 1026 | 1216 | 475 | 3040 | |||
| 5 | 51 | 1020 | 4590 | 8160 | 6375 | 1836 | 22032 | ||
| 6 | 141 | 3384 | 19035 | 45120 | 52875 | 30456 | 6909 | 157920 | |
| 7 | 393 | 11004 | 74277 | 220080 | 343875 | 297108 | 134799 | 25152 | 1106688 |
Compute more
?
Technical details
- Node id
n-9c8afa4b9d121b03· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-d42e6339eab21b9c
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: 1, 4; 2: 3, 24, 27; 3: 7, 84, 189, 112; 4: 19, 304, 1026, 1216, 475; 5: 51, 1020, 4590, 8160, 6375, 1836; 6: 141, 3384, 19035, 45120, 52875, 30456, 6909; 7: 393, 11004, 74277, 220080, 343875, 297108, 134799, 25152; 8: 1107, 35424, 278964, 991872, 1937250, 2231712, 1518804, 566784, 89667; 9: 3139, 113004, 1017036, 4218816, 9887850, 14238504, 12920124, 7232256, 2288331, 313900;
%F FormulaThe row polynomials satisfy (n+(6*n^2-5*n)*x+(11*n^3-17*n^2+5*n)*x^2+(6*n^4-11*n^3+5*n^2-n)*x^3+(n^5-2*n^4+n^3)*x^4)*P(n,x) = (3*n-3+(18*n^2-9*n-9)*x+(33*n^3+18*n^2-45*n-6)*x^2+(18*n^4+69*n^3-36*n^2-57*n+6)*x^3+(3*n^5+39*n^4+36*n^3-57*n^2-30*n+9)*x^4+(6*n^5+24*n^4-3*n^3-24*n^2-6*n+3)*x^5+(3*n^5+3*n^4-3*n^3-3*n^2)*x^6)*P(n-2,x) + (2*n-1+(12*n^2-8*n+1)*x+(22*n^3-11*n^2-16*n+8)*x^2+(12*n^4+12*n^3-33*n^2-4*n+8)*x^3+(2*n^5+11*n^4-14*n^3-8*n^2+4*n+1)*x^4+(2*n^5-n^4-4*n^3+2*n^2+2*n-1)*x^5)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((3*n - 3 + (18*n^2 - 9*n - 9)*x + (33*n^3 + 18*n^2 - 45*n - 6)*x^2 + (18*n^4 + 69*n^3 - 36*n^2 - 57*n + 6)*x^3 + (3*n^5 + 39*n^4 + 36*n^3 - 57*n^2 - 30*n + 9)*x^4 + (6*n^5 + 24*n^4 - 3*n^3 - 24*n^2 - 6*n + 3)*x^5 + (3*n^5 + 3*n^4 - 3*n^3 - 3*n^2)*x^6)*P[n - 2, x] + (2*n - 1 + (12*n^2 - 8*n + 1)*x + (22*n^3 - 11*n^2 - 16*n + 8)*x^2 + (12*n^4 + 12*n^3 - 33*n^2 - 4*n + 8)*x^3 + (2*n^5 + 11*n^4 - 14*n^3 - 8*n^2 + 4*n + 1)*x^4 + (2*n^5 - n^4 - 4*n^3 + 2*n^2 + 2*n - 1)*x^5)*P[n - 1, x])/(n + (6*n^2 - 5*n)*x + (11*n^3 - 17*n^2 + 5*n)*x^2 + (6*n^4 - 11*n^3 + 5*n^2 - n)*x^3 + (n^5 - 2*n^4 + n^3)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((3*n - 3 + (18*n^2 - 9*n - 9)*x + (33*n^3 + 18*n^2 - 45*n - 6)*x^2 + (18*n^4 + 69*n^3 - 36*n^2 - 57*n + 6)*x^3 + (3*n^5 + 39*n^4 + 36*n^3 - 57*n^2 - 30*n + 9)*x^4 + (6*n^5 + 24*n^4 - 3*n^3 - 24*n^2 - 6*n + 3)*x^5 + (3*n^5 + 3*n^4 - 3*n^3 - 3*n^2)*x^6)*v[i - 2] + (2*n - 1 + (12*n^2 - 8*n + 1)*x + (22*n^3 - 11*n^2 - 16*n + 8)*x^2 + (12*n^4 + 12*n^3 - 33*n^2 - 4*n + 8)*x^3 + (2*n^5 + 11*n^4 - 14*n^3 - 8*n^2 + 4*n + 1)*x^4 + (2*n^5 - n^4 - 4*n^3 + 2*n^2 + 2*n - 1)*x^5)*v[i - 1])/(n + (6*n^2 - 5*n)*x + (11*n^3 - 17*n^2 + 5*n)*x^2 + (6*n^4 - 11*n^3 + 5*n^2 - n)*x^3 + (n^5 - 2*n^4 + n^3)*x^4)); 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]
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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
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Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Proof chains ? JSON
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Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
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from
T(n, k) = binomial(n, k)*hypergeom([(1 - n)/2, -n/2], [1], 4).?
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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
T(n, k) = binomial(n, k)*hypergeom([(1 - n)/2, -n/2], [1], 4).?
Related ?
Same fingerprint ?
- A038208 → (θ+1)²
- A003506 → D → (θ+1)²
- A356546 → (θ+1)²
- A178821 → (θ+1)²
- basis-map coefficients of A113214, shifted → (θ+1)²
- A038338 → (θ+1)²
- A038330 → basis A013611 → (θ+1)²
- A028326 → D → (θ+1)²
- A038338 → basis A013609 → (θ+1)²
- basis-map coefficients of A026729, shifted → basis A007318 → (θ+1)²
- A178822 → (θ+1)²
- A124860 → (θ+1)²
- A028326 → basis A013609 → (θ+1)²
- A178821 → basis A013609 → (θ+1)²
- A085880 → D → (θ+1)²
- A178820 → (θ+1)²
- A038325 → (θ+1)²
- A196347 → basis A013609 → (θ+1)²
- basis-map coefficients of A113214, shifted → basis A013609 → (θ+1)²
- A344565 → (θ+1)²
- A135838 → (θ+1)²
- A156992 → basis A013609 → (θ+1)²
- A038208 → D → (θ+1)²
- A085881 → (θ+1)²
- A085880 → basis A013609 → (θ+1)²
- A356546 → basis A013609 → (θ+1)²
- A085881 → basis A013609 → (θ+1)²
- A038221 → basis A013609 → (θ+1)²
- A135065 → (θ+1)²
- A125092: Triangle read by rows: T(n,k) = (k+1)^2*binomial(n,k) (0 <= k <= n).
- A196347 → (θ+1)²
- A028326 → (θ+1)²
- basis-map coefficients of A026729 → basis A007318 → (θ+1)²
- A178822 → basis A013609 → (θ+1)²
- A125092 → reverse
- A038221 → (θ+1)²
- A178820 → basis A013609 → (θ+1)²
- A085880 → (θ+1)²
- A027465 → basis A013611 → (θ+1)²
- A134346 → (θ+1)²
- basis-map coefficients of A112227 → x ↦ x+1 → (θ+1)²
- basis-map coefficients of A127647 → (θ+1)²
- A003506 → (θ+1)²
- A038325 → basis A013609 → (θ+1)²
- basis-map coefficients of A113214, shifted → D → (θ+1)²
- A131431 → x ↦ x+1 → (θ+1)²
- A125092 → reverse → D
- A156992 → (θ+1)²
- A120910 → basis A013610 → (θ+1)²
References
- [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.