{"id":"n-b9d085565acadb7d","revision":0,"first_index":0,"fingerprint":"f6-bdcd1d522b684b55","name":"basis_map_A013611 of A Motzkin related triangle.","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"],["1","4"],["2","12","16"],["4","36","96","64"],["9","104","416","640","256"],["21","300","1600","3840","3840","1024"],["51","864","5760","18880","30720","21504","4096"],["127","2492","19936","82880","188160","222208","114688","16384"],["323","7200","67200","338688","989184","1662976","1490944","589824","65536"],["835","20844","222336","1317120","4709376","10321920","13418496","9437184","2949120","262144"],["2188","60460","725520","4940160","20912640","56641536","97198080","100761600","57016320","14417920","1048576"],["5798","175692","2342560","18025920","88197120","284798976","609337344","843448320","713687040","331612160","69206016","4194304"],["15511","511416","7500768","64366720","357477120","1341407232","3444621312","5994971136","6844907520","4815585280","1868562432","327155712","16777216"],["41835","1490996","23855936","225871360","1403953408","6005615616","18014650368","37870829568","54796025856","52518715392","31188844544","10250878976","1526726656","67108864"],["113634","4353132","75454288","781347840","5374457088","25820354560","88660475904","218811580416","384753008640","470756622336","384162594816","195039330304","54962159616","7046430720","268435456"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","30","200","1425","10625","81875","646875","5211875","42659375","353725000","2965031250","25083859375","213894609375","1836516718750"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","4"]],"recurrence":{"denominator":{"coeffs":[["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["-3","-8","16"],["3","8","-16"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","4"],["2","8"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":204430,"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-0b4a61b492c5d246"]},{"id":204431,"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-0b4a61b492c5d246"]},{"id":204432,"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-0b4a61b492c5d246"]}],"region_exact":{},"incoming_edges":[{"id":"e-0b4a61b492c5d246","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-b9d085565acadb7d","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"3aafae181435c50b91f049c9ae238e39e552b8bd032649ccb50a1cbf83a67ce4","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-de4284a2d5d7d0a6","role":"sequence","name":"A Motzkin related triangle.","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:   1,     4;\n  2:   2,    12,     16;\n  3:   4,    36,     96,      64;\n  4:   9,   104,    416,     640,     256;\n  5:  21,   300,   1600,    3840,    3840,     1024;\n  6:  51,   864,   5760,   18880,   30720,    21504,     4096;\n  7: 127,  2492,  19936,   82880,  188160,   222208,   114688,   16384;\n  8: 323,  7200,  67200,  338688,  989184,  1662976,  1490944,  589824,   65536;\n  9: 835, 20844, 222336, 1317120, 4709376, 10321920, 13418496, 9437184, 2949120, 262144;\n","recurrence_sentence":"The row polynomials satisfy (n+2)*P(n,x) = (3*n-3+(8*n-8)*x-(16*n-16)*x^2)*P(n-2,x) + (2*n+1+(8*n+4)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[((3*n - 3 + (8*n - 8)*x - (16*n - 16)*x^2)*P[n - 2, x] + (2*n + 1 + (8*n + 4)*x)*P[n - 1, x])/(n + 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((3*n - 3 + (8*n - 8)*x - (16*n - 16)*x^2)*v[i - 2] + (2*n + 1 + (8*n + 4)*x)*v[i - 1])/(n + 2)); 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-de4284a2d5d7d0a6","steps":["basis A013611"],"edges":["e-0b4a61b492c5d246"],"text":"A107131 \u2192 basis A013611"},"headline":"A107131 \u2192 basis A013611","chain":null}