A093563: (6,1)-Pascal triangle.

Matches ?

• A093563 triangle 71 terms, 11 rows · tabl · nonn · catalog b-file?

Row sums ? A005009, A135092, A125176

Submission text ?

Triangle begins:
  0: 1;
  1: 6,  1;
  2: 6,  7,   1;
  3: 6, 13,   8,   1;
  4: 6, 19,  21,   9,   1;
  5: 6, 25,  40,  30,  10,   1;
  6: 6, 31,  65,  70,  40,  11,   1;
  7: 6, 37,  96, 135, 110,  51,  12,  1;
  8: 6, 43, 133, 231, 245, 161,  63, 13,  1;
  9: 6, 49, 176, 364, 476, 406, 224, 76, 14, 1;

The row polynomials satisfy (n-1)*P(n,x) = (n-1+(n-1)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + x.

P[0, x_] := 1;
P[1, x_] := 6 + x;
P[n_, x_] := P[n, x] = Expand[Together[((n - 1 + (n - 1)*x)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 6 + x; for(n = 2, N, my(i = n + 1); v[i] = ((n - 1 + (n - 1)*x)*v[i - 1])/(n - 1)); 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)

Triangle begins:
  0: 1;
  1: 6,  1;
  2: 6,  7,   1;
  3: 6, 13,   8,   1;
  4: 6, 19,  21,   9,   1;
  5: 6, 25,  40,  30,  10,   1;
  6: 6, 31,  65,  70,  40,  11,   1;
  7: 6, 37,  96, 135, 110,  51,  12,  1;
  8: 6, 43, 133, 231, 245, 161,  63, 13,  1;
  9: 6, 49, 176, 364, 476, 406, 224, 76, 14, 1;

The row polynomials satisfy (n-1)*P(n,x) = (n-1+(n-1)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + x.

P[0, x_] := 1;
P[1, x_] := 6 + x;
P[n_, x_] := P[n, x] = Expand[Together[((n - 1 + (n - 1)*x)*P[n - 1, x])/(n - 1)]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 6 + x; for(n = 2, N, my(i = n + 1); v[i] = ((n - 1 + (n - 1)*x)*v[i - 1])/(n - 1)); 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(9)

Properties

• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0)
  by Liu–Wang theorem, Row-hypergeometric rows (multiplier sequences)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Liu–Wang theorem, Row-hypergeometric rows (multiplier sequences), Iterated real-rootedness preservers
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Liu–Wang theorem, Row-hypergeometric rows (multiplier sequences), Iterated real-rootedness preservers
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 2
  by Liu–Wang theorem
• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
  by Iterated real-rootedness preservers
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 1
  by Iterated real-rootedness preservers

Proof chains ? JSON

• Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
• 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
• Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Used in 5

• basis_map_A013609 of (6,1)-Pascal triangle. Image under a basis map u^k -> P_k (A013609)
• basis_map_A013610 of (6,1)-Pascal triangle. Image under a basis map u^k -> P_k (A013610)
• basis_map_A007318 of (6,1)-Pascal triangle. Image under a basis map u^k -> P_k (A007318)
• basis_map_A008277 of (6,1)-Pascal triangle. Image under a basis map u^k -> P_k (A008277)
• basis_map_A013611 of (6,1)-Pascal triangle. Image under a basis map u^k -> P_k (A013611)

Related ?

Same fingerprint ?

• A096956: Pascal (1,6) triangle.

References

1. [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
2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
3. [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
4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.