{"id":"n-9e0feeda3da88095","revision":0,"first_index":0,"fingerprint":"f6-6a0276fa01fb1cc5","name":"basis_map_A008277 of The (1,2)-Pascal triangle (or Lucas triangle) read by rows.","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":[["2"],["3","2"],["6","9","2"],["12","33","17","2"],["24","111","101","27","2"],["48","357","515","236","39","2"],["96","1119","2417","1695","470","53","2"],["192","3453","10787","10892","4515","841","69","2"],["384","10551","46601","65247","37982","10402","1393","87","2"],["768","32037","196955","372836","293139","110796","21546","2176","107","2"],["1536","96879","819857","2061135","2131670","1068711","283164","41130","3246","129","2"],["3072","292173","3376307","11125532","14851155","9612647","3334023","653334","73590","4665","153","2"],["6144","879591","13797401","59003967","100232462","82139684","36284831","9214029","1389234","124905","6501","179","2"],["12288","2644917","56069195","308817236","660398739","675210250","372418332","119211092","23106369","2763189","202917","8828","207","2"],["24576","7947039","226921697","1600155375","4271209670","5386870489","3654556906","1445318160","350274782","53501448","5198193","317681","11726","237","2"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["2","5","17","64","265","1197","5852","30751","172649","1030362","6507459","43326653","303078930","2220863561","17002307981"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["2"],["3","2"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["-2","-1"],["2","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-1"],["0","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":195940,"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-85f08c5fc83cf41e"]},{"id":195941,"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-85f08c5fc83cf41e"]},{"id":195942,"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-85f08c5fc83cf41e"]}],"region_exact":{},"incoming_edges":[{"id":"e-85f08c5fc83cf41e","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-9e0feeda3da88095","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":"9792a39decb8dd17d33238eda2a10e9ef5ce58cf9ed1b2deafb326453f312cd3","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-b636fb6adf08770c","role":"sequence","name":"The (1,2)-Pascal triangle (or Lucas triangle) read by rows.","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:   2;\n  1:   3,     2;\n  2:   6,     9,      2;\n  3:  12,    33,     17,      2;\n  4:  24,   111,    101,     27,      2;\n  5:  48,   357,    515,    236,     39,      2;\n  6:  96,  1119,   2417,   1695,    470,     53,     2;\n  7: 192,  3453,  10787,  10892,   4515,    841,    69,    2;\n  8: 384, 10551,  46601,  65247,  37982,  10402,  1393,   87,   2;\n  9: 768, 32037, 196955, 372836, 293139, 110796, 21546, 2176, 107, 2;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (2*n-2+(n-1)*x)*P(n-1,x) + (n-1)*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2, P(1,x) = 3 + 2*x.","mathematica":"P[0, x_] := 2;\nP[1, x_] := 3 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n - 2 + (n - 1)*x)*P[n - 1, x] + (n - 1)*x*D[P[n - 1, x], x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 2 + (n - 1)*x)*v[i - 1] + (n - 1)*x*deriv(v[i - 1], x))/(n - 1)); 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-b636fb6adf08770c","steps":["basis A008277"],"edges":["e-85f08c5fc83cf41e"],"text":"A029635 \u2192 basis A008277"},"headline":"A029635 \u2192 basis A008277","chain":null}