{"id":"n-78db5665b18c691b","revision":1,"first_index":0,"fingerprint":"f6-e5461fde6b0fcf98","name":"basis_map_A008277 of Triangular array binomial(2*n-k, k), k=0..n, n >= 0.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2","1"],["5","6","1"],["13","30","12","1"],["34","137","100","20","1"],["89","594","703","250","30","1"],["233","2498","4490","2499","525","42","1"],["610","10313","27040","21956","7070","980","56","1"],["1597","42094","156773","177772","81576","17094","1680","72","1"],["4181","170582","886064","1362605","851486","250368","36834","2700","90","1"],["10946","688105","4920588","10051716","8295435","3267264","667128","72666","4125","110","1"],["28657","2767546","26988659","72131910","76916558","39323103","10630312","1593900","133683","6050","132","1"],["75025","11109786","146718942","507204831","687574051","446110450","154641747","30432116","3491202","232375","8580","156","1"],["196418","44542825","792487848","3512482644","5978634424","4840738196","2102844920","525842083","78642135","7123402","385385","11830","182","1"],["514229","178442742","4260406153","24043899816","50889030256","50756369804","27159694688","8431613172","1591109650","186873258","13701402","614341","15925","210","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","12","56","292","1667","10288","68026","478659","3564911","27978084","230520511","1987599262","17883932293","167512285647"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","-3","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-3","-2"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1"]]},"deriv_order":2,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162195,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-2066319958628df3"]},{"id":162196,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-2066319958628df3"]},{"id":162197,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-2066319958628df3"]}],"region_exact":{},"incoming_edges":[{"id":"e-2066319958628df3","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-78db5665b18c691b","premise_count":1,"theorem":"ray basis maps: for L f = (a + b t) f + kappa (t - h) f', b > 0, kappa >= 0, and g real-rooted with every root <= h, of multiplicity m at h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in (-inf, h] if and only if S lies in (-inf, a + b h + kappa m] (RealRooted rayBasisMap_maps_realRootedIn_iff; the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"841d0e9bd65b20e6b9e5c4a82e5c64b281d6e9fdbd6db3e0f06da28adde72402","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-6758fe7729845be8","role":"sequence","name":"Triangular array binomial(2*n-k, k), k=0..n, n >= 0.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:    1;\n  1:    2,      1;\n  2:    5,      6,      1;\n  3:   13,     30,     12,       1;\n  4:   34,    137,    100,      20,      1;\n  5:   89,    594,    703,     250,     30,      1;\n  6:  233,   2498,   4490,    2499,    525,     42,     1;\n  7:  610,  10313,  27040,   21956,   7070,    980,    56,    1;\n  8: 1597,  42094, 156773,  177772,  81576,  17094,  1680,   72,  1;\n  9: 4181, 170582, 886064, 1362605, 851486, 250368, 36834, 2700, 90, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(1+3*x+x^2)*P(n-2,x) - (3*x+2*x^2)*d/dx P(n-2,x) - x^2*d^2/dx^2 P(n-2,x) + (3+2*x)*P(n-1,x) + 2*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[-(1 + 3*x + x^2)*P[n - 2, x] - (3*x + 2*x^2)*D[P[n - 2, x], x] - x^2*D[P[n - 2, x], {x, 2}] + (3 + 2*x)*P[n - 1, x] + 2*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = -(1 + 3*x + x^2)*v[i - 2] - (3*x + 2*x^2)*deriv(v[i - 2], x) - x^2*deriv(deriv(v[i - 2], x), x) + (3 + 2*x)*v[i - 1] + 2*x*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-6758fe7729845be8","steps":["basis A008277"],"edges":["e-2066319958628df3"],"text":"A054142 \u2192 basis A008277"},"headline":"A054142 \u2192 basis A008277","chain":null}
