A078812: Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).

OEIS matches ?

• A078812: triangle, 66 terms agree, 11 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?
• A001906: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A088305: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A172431: reversed triangle, 55 terms agree, 10 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 1 ?
• A237268: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

Triangle begins:
   1:  1;
   2:  2,   1;
   3:  3,   4,   1;
   4:  4,  10,   6,    1;
   5:  5,  20,  21,    8,    1;
   6:  6,  35,  56,   36,   10,    1;
   7:  7,  56, 126,  120,   55,   12,   1;
   8:  8,  84, 252,  330,  220,   78,  14,   1;
   9:  9, 120, 462,  792,  715,  364, 105,  16,  1;
  10: 10, 165, 792, 1716, 2002, 1365, 560, 136, 18, 1;

The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) - P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + x.

P[1, x_] := 1;
P[2, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[(2 + x)*P[n - 1, x] - P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + x; for(n = 3, N, my(i = n); v[i] = (2 + x)*v[i - 1] - v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Triangle begins:
  0:  1;
  1:  2,   1;
  2:  3,   4,   1;
  3:  4,  10,   6,    1;
  4:  5,  20,  21,    8,    1;
  5:  6,  35,  56,   36,   10,    1;
  6:  7,  56, 126,  120,   55,   12,   1;
  7:  8,  84, 252,  330,  220,   78,  14,   1;
  8:  9, 120, 462,  792,  715,  364, 105,  16,  1;
  9: 10, 165, 792, 1716, 2002, 1365, 560, 136, 18, 1;

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

P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[(2 + x)*P[n - 1, x] - P[n - 2, x]];
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] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (2 + x)*v[i - 1] - v[i - 2]); 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)

Triangle begins:
   1: 1;
   2: 1,  2;
   3: 1,  4,   3;
   4: 1,  6,  10,   4;
   5: 1,  8,  21,  20,    5;
   6: 1, 10,  36,  56,   35,    6;
   7: 1, 12,  55, 120,  126,   56,    7;
   8: 1, 14,  78, 220,  330,  252,   84,   8;
   9: 1, 16, 105, 364,  715,  792,  462, 120,   9;
  10: 1, 18, 136, 560, 1365, 2002, 1716, 792, 165, 10;

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (2+x)*P(n-1,x) - P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + x.

P[1, x_] := 1;
P[2, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[(2 + x)*P[n - 1, x] - P[n - 2, x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + x; for(n = 3, N, my(i = n); v[i] = (2 + x)*v[i - 1] - v[i - 2]); 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)

Proved properties

• Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-1502106ad4631c34
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m >= 2
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-1502106ad4631c34, e-537ef91063de36e5, e-5aa1c8cf88fab58c
• Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0)
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-537ef91063de36e5
• Coefficients of one sign. Every nonzero row, n >= 1: the coefficients of P_n have one sign
  All coefficients of each polynomial have the same sign (after x -> -x when alternating).
  derived from the cited steps by a stated theorem: e-537ef91063de36e5, e-5aa1c8cf88fab58c
• Hadamard kernel (Malo). Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)
  Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
  derived from the cited steps by a stated theorem: e-537ef91063de36e5, e-5aa1c8cf88fab58c
• Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in (-4, 0)
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-5aa1c8cf88fab58c
• Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in [-1, 1] to roots in (-inf, +inf): every real polynomial with all roots in E = [-1, 1] has a real-rooted image
  The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.
  by the cited steps: e-5a12e548117a4094
• Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in [-1, 1] to roots in [-4, 0]: every real polynomial with all roots in E = [-1, 1] has an image with all roots in I = [-4, 0]
  The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.
  by the cited steps: e-5a12e548117a4094
• Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in [-1, 0] to roots in [-4, 0]: every PF input with all roots in [-1, 0] has a PF image; no larger [-R, 0] works
  The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.
  by the cited steps: e-5a12e548117a4094

Proof chains

4 incoming proof steps. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• basis_map_theorem e-5a12e548117a4094 root: theorem check

  Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

• theorem_check three_term_recurrence e-1502106ad4631c34 root: theorem check

  Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

• theorem_check three_term_recurrence e-537ef91063de36e5 root: theorem check

  Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

• theorem_check three_term_recurrence e-5aa1c8cf88fab58c root: theorem check

  Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Used in 14 proof steps

• e-13742d707ef3ea03 basis_map (A008277) → basis_map_A008277 of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).
• e-7d8fca0f88bff382 basis_map (A013611) → basis_map_A013611 of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).
• e-95c725304ba970c4 basis_map (A007318) → basis_map_A007318 of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).
• e-a4eb33b32804b1a0 basis_map (A013610) → basis_map_A013610 of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).
• e-f19db33e28acf0c7 basis_map (A013609) → basis_map_A013609 of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).
• e-1dbefba2139b076a preserver_image (derivative_polynomial, shift 0) → derivative of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).
• e-2147a0005767de5d preserver_image (euler_polynomial, shift 0) → theta_plus_one of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).
• e-32f4efbf40925a37 preserver_image (reversal, shift 0) → reversal of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).
• e-67a2e8580e66d57b preserver_image (euler_polynomial, shift 0) → theta_plus_one_squared of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).
• e-7a38528e1fbfc148 preserver_image (derivative_polynomial, shift 0) → one_plus_derivative of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).
• e-c92b2c850587f23b preserver_image (mobius, shift 0) → mobius_x_over_x_plus_one of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).
• e-f4c2062c2af62bf3 preserver_image (affine, shift 0) → basis_map_A007318 of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).
• e-40caf6cac4d83759 veronese_section (r=2, j=1, shift 0) → veronese_section_2_1 of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).
• e-b8f275a63993d6a1 veronese_section (r=2, j=0, shift 0) → veronese_section_2_0 of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).

Related sequences ?

Same fingerprint (5)

Equal up to content, sign, reversal and a small index shift.

• A172431: Even row Pascal-square read by antidiagonals.
• A172431 → reverse
• A078812 → reverse
• A082985 → D
• A091044 → basis A013611

References

1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
2. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023