A109983: Triangle read by rows: T(n, k) (0<=k<=2n) is the number of Delannoy paths of length n, having k steps.
A109983 ↗ nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence from A109983
\(\left(n\right)\,P_{n}(x) = \left(-x - 2\,x^{2} + 2\,n x + 4\,n x^{2}\right)\,P_{n-1}(x) + \left(x^{2} - n x^{2}\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = x + 2\,x^{2}\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | ||||||||||||||
| 1 | 0 | 1 | 2 | 3 | ||||||||||||
| 2 | 0 | 0 | 1 | 6 | 6 | 13 | ||||||||||
| 3 | 0 | 0 | 0 | 1 | 12 | 30 | 20 | 63 | ||||||||
| 4 | 0 | 0 | 0 | 0 | 1 | 20 | 90 | 140 | 70 | 321 | ||||||
| 5 | 0 | 0 | 0 | 0 | 0 | 1 | 30 | 210 | 560 | 630 | 252 | 1683 | ||||
| 6 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 42 | 420 | 1680 | 3150 | 2772 | 924 | 8989 | ||
| 7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 56 | 756 | 4200 | 11550 | 16632 | 12012 | 3432 | 48639 |
All 15 rows b-files: A001850 (row sums), A063007, A104684 (reversed triangle), A109983 ?
Compute more
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Technical details
- Node id
n-e4d248920a317259· revision 1 · JSON- Definition source
oeis_catalog:A109983- Fingerprint
f6-fd15deff59829444
Observed on the stored rows ?
- Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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 ?
- A063007 triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?
- A104684 reversed triangle 50 terms, 9 rows · tabl · nonn · catalog b-file?
- A109983 triangle 64 terms, 8 rows · tabf · nonn · catalog b-file?
Row sums ? A001850
Submission text ?
Atlas rows n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 1; 1: 0, 1, 2; 2: 0, 0, 1, 6, 6; 3: 0, 0, 0, 1, 12, 30, 20; 4: 0, 0, 0, 0, 1, 20, 90, 140, 70; 5: 0, 0, 0, 0, 0, 1, 30, 210, 560, 630, 252; 6: 0, 0, 0, 0, 0, 0, 1, 42, 420, 1680, 3150, 2772, 924; 7: 0, 0, 0, 0, 0, 0, 0, 1, 56, 756, 4200, 11550, 16632, 12012, 3432; 8: 0, 0, 0, 0, 0, 0, 0, 0, 1, 72, 1260, 9240, 34650, 72072, 84084, 51480, 12870; 9: 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 90, 1980, 18480, 90090, 252252, 420420, 411840, 218790, 48620;
%F FormulaThe row polynomials satisfy n*P(n,x) = ((2*n-1)*x+(4*n-2)*x^2)*P(n-1,x) - (n-1)*x^2*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x + 2*x^2.
%t MathematicaP[0, x_] := 1;
P[1, x_] := x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(((2*n - 1)*x + (4*n - 2)*x^2)*P[n - 1, x] - (n - 1)*x^2*P[n - 2, x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = x + 2*x^2; for(n = 2, N, my(i = n + 1); v[i] = (((2*n - 1)*x + (4*n - 2)*x^2)*v[i - 1] - (n - 1)*x^2*v[i - 2])/(n)); 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
A063007 triangle, offset 0, from P0, low zeros dropped
%e ExampleTriangle begins: 0: 1; 1: 1, 2; 2: 1, 6, 6; 3: 1, 12, 30, 20; 4: 1, 20, 90, 140, 70; 5: 1, 30, 210, 560, 630, 252; 6: 1, 42, 420, 1680, 3150, 2772, 924; 7: 1, 56, 756, 4200, 11550, 16632, 12012, 3432; 8: 1, 72, 1260, 9240, 34650, 72072, 84084, 51480, 12870; 9: 1, 90, 1980, 18480, 90090, 252252, 420420, 411840, 218790, 48620;
%F FormulaRow n lists the coefficients of P(n,x) in ascending powers of x, starting at the lowest nonzero coefficient, where n*P(n,x) = ((2*n-1)*x+(4*n-2)*x^2)*P(n-1,x) - (n-1)*x^2*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x + 2*x^2.
%t MathematicaP[0, x_] := 1;
P[1, x_] := x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(((2*n - 1)*x + (4*n - 2)*x^2)*P[n - 1, x] - (n - 1)*x^2*P[n - 2, x])/(n)]];
Table[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n + 1], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = x + 2*x^2; for(n = 2, N, my(i = n + 1); v[i] = (((2*n - 1)*x + (4*n - 2)*x^2)*v[i - 1] - (n - 1)*x^2*v[i - 2])/(n)); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c = concat(c, vector(n + 1 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)polytool c48629374597
A104684 reversed triangle, offset 0, from P0, low zeros dropped
%e ExampleTriangle begins: 0: 1; 1: 2, 1; 2: 6, 6, 1; 3: 20, 30, 12, 1; 4: 70, 140, 90, 20, 1; 5: 252, 630, 560, 210, 30, 1; 6: 924, 2772, 3150, 1680, 420, 42, 1; 7: 3432, 12012, 16632, 11550, 4200, 756, 56, 1; 8: 12870, 51480, 84084, 72072, 34650, 9240, 1260, 72, 1; 9: 48620, 218790, 411840, 420420, 252252, 90090, 18480, 1980, 90, 1;
%F FormulaRow n lists the coefficients of P(n,x) in descending powers of x, down to the lowest nonzero coefficient, where n*P(n,x) = ((2*n-1)*x+(4*n-2)*x^2)*P(n-1,x) - (n-1)*x^2*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x + 2*x^2.
%t MathematicaP[0, x_] := 1;
P[1, x_] := x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(((2*n - 1)*x + (4*n - 2)*x^2)*P[n - 1, x] - (n - 1)*x^2*P[n - 2, x])/(n)]];
Table[Reverse[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n + 1]], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = x + 2*x^2; for(n = 2, N, my(i = n + 1); v[i] = (((2*n - 1)*x + (4*n - 2)*x^2)*v[i - 1] - (n - 1)*x^2*v[i - 2])/(n)); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)polytool c48629374597
A109983 triangle, offset 0, from P0
%e ExampleTriangle begins: 0: 1; 1: 0, 1, 2; 2: 0, 0, 1, 6, 6; 3: 0, 0, 0, 1, 12, 30, 20; 4: 0, 0, 0, 0, 1, 20, 90, 140, 70; 5: 0, 0, 0, 0, 0, 1, 30, 210, 560, 630, 252; 6: 0, 0, 0, 0, 0, 0, 1, 42, 420, 1680, 3150, 2772, 924; 7: 0, 0, 0, 0, 0, 0, 0, 1, 56, 756, 4200, 11550, 16632, 12012, 3432; 8: 0, 0, 0, 0, 0, 0, 0, 0, 1, 72, 1260, 9240, 34650, 72072, 84084, 51480, 12870; 9: 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 90, 1980, 18480, 90090, 252252, 420420, 411840, 218790, 48620;
%F FormulaThe row polynomials satisfy n*P(n,x) = ((2*n-1)*x+(4*n-2)*x^2)*P(n-1,x) - (n-1)*x^2*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x + 2*x^2.
%t MathematicaP[0, x_] := 1;
P[1, x_] := x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(((2*n - 1)*x + (4*n - 2)*x^2)*P[n - 1, x] - (n - 1)*x^2*P[n - 2, x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = x + 2*x^2; for(n = 2, N, my(i = n + 1); v[i] = (((2*n - 1)*x + (4*n - 2)*x^2)*v[i - 1] - (n - 1)*x^2*v[i - 2])/(n)); 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
- Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
Proof chains ? JSON
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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
Used in 12
- shift_x_plus_one of Triangle read by rows: T(n, k) (0<=k<=2n) is the number of … Image under a basis map u^k -> P_k (A007318)
- basis_map_A013610 of Triangle read by rows: T(n, k) (0<=k<=2n) is the number of… Image under a basis map u^k -> P_k (A013610)
- basis_map_A013609 of Triangle read by rows: T(n, k) (0<=k<=2n) is the number of… Image under a basis map u^k -> P_k (A013609)
- basis_map_A008277 of Triangle read by rows: T(n, k) (0<=k<=2n) is the number of… Image under a basis map u^k -> P_k (A008277)
- basis_map_A013611 of Triangle read by rows: T(n, k) (0<=k<=2n) is the number of… Image under a basis map u^k -> P_k (A013611)
- theta_plus_one_squared of Triangle read by rows: T(n, k) (0<=k<=2n) is the numb… Image under a real-rootedness preserver (euler polynomial, shift 0)
- mobius_x_over_x_plus_one of Triangle read by rows: T(n, k) (0<=k<=2n) is the nu… Image under a real-rootedness preserver (mobius, shift 0)
- derivative of Triangle read by rows: T(n, k) (0<=k<=2n) is the number of Delann… Image under a real-rootedness preserver (derivative polynomial, shift 0)
4 more
- one_plus_derivative of Triangle read by rows: T(n, k) (0<=k<=2n) is the number … Image under a real-rootedness preserver (derivative polynomial, shift 0)
- shift_x_plus_one of Triangle read by rows: T(n, k) (0<=k<=2n) is the number of … Image under a real-rootedness preserver (affine, shift 0)
- theta_plus_one of Triangle read by rows: T(n, k) (0<=k<=2n) is the number of De… Image under a real-rootedness preserver (euler polynomial, shift 0)
- basis_map_A007318 of Square the entries of Pascal's triangle. Image under a real-rootedness preserver (reversal, shift 0)
Related ?
Same fingerprint ?
- A104684: Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) …
- A063007: T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.
- A069466 → basis A007318 → reverse
- A069466 → basis A007318
- A253284 → θ+1
- A008459 → x ↦ x/(x+1)
- A008459 → x ↦ x+1
- A069466 → x ↦ x/(x+1)
- A008459 → x ↦ x+1 → reverse
References
- [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.