A036969: Triangle read by rows: T(n,k) = T(n-1,k-1) + k^2*T(n-1,k), 1 < k <= n, T(n,1) = 1.
A036969 ↗ nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence from A036969
\(P_{n}(x) = \left(1 + x\right)\,P_{n-1}(x) + \left(3\,x\right)\,P_{n-1}'(x) + \left(x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | ||||||||||||||
| 1 | 1 | 1 | 2 | |||||||||||||
| 2 | 1 | 5 | 1 | 7 | ||||||||||||
| 3 | 1 | 21 | 14 | 1 | 37 | |||||||||||
| 4 | 1 | 85 | 147 | 30 | 1 | 264 | ||||||||||
| 5 | 1 | 341 | 1408 | 627 | 55 | 1 | 2433 | |||||||||
| 6 | 1 | 1365 | 13013 | 11440 | 2002 | 91 | 1 | 27913 | ||||||||
| 7 | 1 | 5461 | 118482 | 196053 | 61490 | 5278 | 140 | 1 | 386906 | |||||||
| 8 | 1 | 21845 | 1071799 | 3255330 | 1733303 | 251498 | 12138 | 204 | 1 | 6346119 | ||||||
| 9 | 1 | 87381 | 9668036 | 53157079 | 46587905 | 10787231 | 846260 | 25194 | 285 | 1 | 121159373 | |||||
| 10 | 1 | 349525 | 87099705 | 860181300 | 1217854704 | 434928221 | 52253971 | 2458676 | 48279 | 385 | 1 | 2655174768 | ||||
| 11 | 1 | 1398101 | 784246870 | 13850000505 | 31306548900 | 16875270660 | 2995372800 | 209609235 | 6369275 | 86779 | 506 | 1 | 66028903633 | |||
| 12 | 1 | 5592405 | 7059619931 | 222384254950 | 796513723005 | 638816292660 | 163648537860 | 16410363840 | 725520510 | 15047175 | 148005 | 650 | 1 | 1845579100993 | ||
| 13 | 1 | 22369621 | 63542171784 | 3565207699131 | 20135227330075 | 23793900258765 | 8657594647800 | 1213911823620 | 75177525150 | 2230238010 | 32955780 | 241605 | 819 | 1 | 57506847262162 | |
| 14 | 1 | 89478485 | 571901915677 | 57106865357880 | 506945890951006 | 876715636645615 | 448016038000965 | 86347951359480 | 7303291360770 | 298201326150 | 6217887390 | 67746900 | 380016 | 1015 | 1 | 1983312152411351 |
b-files: A008957 (reversed triangle), A036969 ?
Compute more
?
Technical details
- Node id
n-c856ca922af03076· revision 1 · JSON- Definition source
oeis_catalog:A036969- Fingerprint
f6-de3a1cd8e58b208d
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 ?
- A036969 triangle 50 terms, 9 rows · tabl · nonn · catalog b-file?
- A008957 reversed triangle 50 terms, 9 rows · tabl · nonn · catalog b-file?
Row sums ? A135920
Submission text ?
Atlas rows n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 1; 1: 1, 1; 2: 1, 5, 1; 3: 1, 21, 14, 1; 4: 1, 85, 147, 30, 1; 5: 1, 341, 1408, 627, 55, 1; 6: 1, 1365, 13013, 11440, 2002, 91, 1; 7: 1, 5461, 118482, 196053, 61490, 5278, 140, 1; 8: 1, 21845, 1071799, 3255330, 1733303, 251498, 12138, 204, 1; 9: 1, 87381, 9668036, 53157079, 46587905, 10787231, 846260, 25194, 285, 1;
%F FormulaThe row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + 3*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[(1 + x)*P[n - 1, x] + 3*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] = (1 + x)*v[i - 1] + 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};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)polytool c48629374597
A008957 reversed triangle, offset 1, from P0
%e ExampleTriangle begins: 1: 1; 2: 1, 1; 3: 1, 5, 1; 4: 1, 14, 21, 1; 5: 1, 30, 147, 85, 1; 6: 1, 55, 627, 1408, 341, 1; 7: 1, 91, 2002, 11440, 13013, 1365, 1; 8: 1, 140, 5278, 61490, 196053, 118482, 5461, 1; 9: 1, 204, 12138, 251498, 1733303, 3255330, 1071799, 21845, 1; 10: 1, 285, 25194, 846260, 10787231, 46587905, 53157079, 9668036, 87381, 1;
%F FormulaRow n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+x)*P(n-1,x) + 3*x*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.
%t MathematicaP[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x] + 3*x*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + x)*v[i - 1] + 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 - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)polytool c48629374597
A036969 triangle, offset 1, from P0
%e ExampleTriangle begins: 1: 1; 2: 1, 1; 3: 1, 5, 1; 4: 1, 21, 14, 1; 5: 1, 85, 147, 30, 1; 6: 1, 341, 1408, 627, 55, 1; 7: 1, 1365, 13013, 11440, 2002, 91, 1; 8: 1, 5461, 118482, 196053, 61490, 5278, 140, 1; 9: 1, 21845, 1071799, 3255330, 1733303, 251498, 12138, 204, 1; 10: 1, 87381, 9668036, 53157079, 46587905, 10787231, 846260, 25194, 285, 1;
%F FormulaThe row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + 3*x*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.
%t MathematicaP[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x] + 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], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + x)*v[i - 1] + 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 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)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
- Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
Proof chains ? JSON
Used in 7
- theta_plus_one_squared of Triangle read by rows: T(n,k) = T(n-1,k-1) + k^2*T(n-… Image under a real-rootedness preserver (euler polynomial, shift 0)
- derivative of Triangle read by rows: T(n,k) = T(n-1,k-1) + k^2*T(n-1,k), 1 < k … Image under a real-rootedness preserver (derivative polynomial, shift 0)
- shift_x_plus_one of Triangle read by rows: T(n,k) = T(n-1,k-1) + k^2*T(n-1,k), … Image under a real-rootedness preserver (affine, shift 0)
- mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) = T(n-1,k-1) + k^2*T(… Image under a real-rootedness preserver (mobius, shift 0)
- one_plus_derivative of Triangle read by rows: T(n,k) = T(n-1,k-1) + k^2*T(n-1,k… Image under a real-rootedness preserver (derivative polynomial, shift 0)
- Triangle of central factorial numbers T(2*n,2*n-2*k), k >= 0, n >= 1 (in Riorda… Image under a real-rootedness preserver (reversal, shift 0)
- theta_plus_one of Triangle read by rows: T(n,k) = T(n-1,k-1) + k^2*T(n-1,k), 1 … Image under a real-rootedness preserver (euler polynomial, shift 0)
Related ?
Same fingerprint ?
- A008957: Triangle of central factorial numbers T(2*n,2*n-2*k), k >= 0, n >= 1 (in Riordan's notation).
- A269945 → reverse
- A269945: Triangle read by rows. Stirling set numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise …
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.