A371081: Triangle read by rows, (2, 2)-Lah numbers.
A371081 ↗ nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence from A371081
\(P_{n}(x) = \left(9 + x + 6\,n + n^{2}\right)\,P_{n-1}(x) + \left(7\,x + 2\,n x\right)\,P_{n-1}'(x) + \left(x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 16 | 1 | 17 | ||||||
| 2 | 400 | 52 | 1 | 453 | |||||
| 3 | 14400 | 2948 | 116 | 1 | 17465 | ||||
| 4 | 705600 | 203072 | 12344 | 216 | 1 | 921233 | |||
| 5 | 45158400 | 17154432 | 1437472 | 38480 | 360 | 1 | 63789145 | ||
| 6 | 3657830400 | 1760601600 | 191088544 | 6978592 | 99320 | 556 | 1 | 5616599013 | |
| 7 | 365783040000 | 216690624000 | 29277351936 | 1370470592 | 26445312 | 224420 | 812 | 1 | 613148157073 |
All 15 rows b-files: A371081, A371488 (row sums) ?
Compute more
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Technical details
- Node id
n-ee73bfb592710e97· revision 0 · JSON- Definition source
oeis_catalog:A371081- Fingerprint
f6-c0f485d760aa07f8
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 ?
Row sums ? A371488
Submission text ?
Atlas rows n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 1; 1: 16, 1; 2: 400, 52, 1; 3: 14400, 2948, 116, 1; 4: 705600, 203072, 12344, 216, 1; 5: 45158400, 17154432, 1437472, 38480, 360, 1; 6: 3657830400, 1760601600, 191088544, 6978592, 99320, 556, 1; 7: 365783040000, 216690624000, 29277351936, 1370470592, 26445312, 224420, 812, 1; 8: 44259747840000, 31569232896000, 5164563101184, 297889587968, 7320665792, 83896832, 459088, 1136, 1; 9: 6373403688960000, 5379460107264000, 1043823600728064, 72189720393984, 2171980030720, 31566850240, 232641344, 869184, 1536, 1;
%F FormulaThe row polynomials satisfy P(n,x) = (n^2+6*n+9+x)*P(n-1,x) + (2*n+7)*x*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.
%t MathematicaP[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(n^2 + 6*n + 9 + x)*P[n - 1, x] + (2*n + 7)*x*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (n^2 + 6*n + 9 + x)*v[i - 1] + (2*n + 7)*x*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), x)); 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
A371081 triangle, offset 2, from P0
%e ExampleTriangle begins: 2: 1; 3: 16, 1; 4: 400, 52, 1; 5: 14400, 2948, 116, 1; 6: 705600, 203072, 12344, 216, 1; 7: 45158400, 17154432, 1437472, 38480, 360, 1; 8: 3657830400, 1760601600, 191088544, 6978592, 99320, 556, 1; 9: 365783040000, 216690624000, 29277351936, 1370470592, 26445312, 224420, 812, 1; 10: 44259747840000, 31569232896000, 5164563101184, 297889587968, 7320665792, 83896832, 459088, 1136, 1; 11: 6373403688960000, 5379460107264000, 1043823600728064, 72189720393984, 2171980030720, 31566850240, 232641344, 869184, 1536, 1;
%F FormulaThe row polynomials satisfy P(n,x) = (n^2+2*n+1+x)*P(n-1,x) + (2*n+3)*x*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 3, with P(2,x) = 1.
%t MathematicaP[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(n^2 + 2*n + 1 + x)*P[n - 1, x] + (2*n + 3)*x*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];
Table[PadRight[CoefficientList[P[n, x], x], n - 1], {n, 2, 11}] // Flatten%o PARI/GPP(N) = {my(v = vector(N - 1)); v[1] = 1; for(n = 3, N, my(i = n - 1); v[i] = (n^2 + 2*n + 1 + x)*v[i - 1] + (2*n + 3)*x*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - 1 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i + 1, v[i])))};
T(11)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 Iterated real-rootedness preservers
-
Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Iterated real-rootedness preservers
Proof chains ? JSON
Used in 7
- shift_x_plus_one of Triangle read by rows, (2, 2)-Lah numbers. Image under a real-rootedness preserver (affine, shift 0)
- derivative of Triangle read by rows, (2, 2)-Lah numbers. Image under a real-rootedness preserver (derivative polynomial, shift 0)
- theta_plus_one of Triangle read by rows, (2, 2)-Lah numbers. Image under a real-rootedness preserver (euler polynomial, shift 0)
- mobius_x_over_x_plus_one of Triangle read by rows, (2, 2)-Lah numbers. Image under a real-rootedness preserver (mobius, shift 0)
- reversal of Triangle read by rows, (2, 2)-Lah numbers. Image under a real-rootedness preserver (reversal, shift 0)
- one_plus_derivative of Triangle read by rows, (2, 2)-Lah numbers. Image under a real-rootedness preserver (derivative polynomial, shift 0)
- theta_plus_one_squared of Triangle read by rows, (2, 2)-Lah numbers. Image under a real-rootedness preserver (euler polynomial, shift 0)
Related ?
Same fingerprint ?
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.