{"id":"n-40234e58e7c7f4e9","revision":0,"first_index":0,"fingerprint":"f6-e5f05be7597b79dc","name":"basis_map_A007318 of Triangular array:  the fusion of polynomial sequences P and Q given by p(n,x)=(x+2)^n and q(n,x)=(x+2)^n.","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":true,"rows":[["1"],["3","2"],["21","29","10"],["147","308","215","50"],["1029","2891","3045","1425","250"],["7203","25382","35770","25200","8875","1250"],["50421","213689","377300","355250","188125","53125","6250"],["352947","1747928","3709545","4373250","3093125","1312500","309375","31250"],["2470629","14000231","34706455","49160475","43518125","24653125","8728125","1765625","156250"],["17294403","110354762","312946340","517655600","550429250","390162500","184362500","56000000","9921875","781250"],["121060821","858955349","2742398190","5188320900","6441282750","5483283750","3241350000","1313812500","349453125","55078125","3906250"],["847425747","6617991548","23491564075","50030237250","71030583750","70589400000","50105868750","25403437500","9015234375","2132812500","302734375","19531250"],["5931980229","50563069571","197530906265","467669481125","747365272500","849278718750","703688081250","428353406250","190123828125","60005859375","12783203125","1650390625","97656250"],["41523861603","383601388142","1635531691710","4261340899200","7569904313125","9681777393750","9172210162500","6516914250000","3472633828125","1370660156250","389511718750","75468750000","8935546875","488281250"],["290667031221","2892829025009","13366728782680","38007044752950","74296034687875","105621963321875","112614358106250","91479450562500","56893008046875","26957790234375","9579882812500","2475839843750","439892578125","48095703125","2441406250"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","60","720","8640","103680","1244160","14929920","179159040","2149908480","25798901760","309586821120","3715041853440","44580502241280","534966026895360"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["3","2"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["-7","-5"],["7","5"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":195583,"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-3c1a0c86d879ec10"]},{"id":195584,"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-3c1a0c86d879ec10"]},{"id":195585,"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-3c1a0c86d879ec10"]},{"id":195586,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-3c1a0c86d879ec10"]}],"region_exact":{},"incoming_edges":[{"id":"e-3c1a0c86d879ec10","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-40234e58e7c7f4e9","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":"6d8d9f9f427d173e9200b2028d436ab81a8405e331e127e63c6d41e36daf4290","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-cedb1276760d9bb1","role":"sequence","name":"Triangular array:  the fusion of polynomial sequences P and Q given by p(n,x)=(x+2)^n and q(n,x)=(x+2)^n.","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:        3,         2;\n  2:       21,        29,        10;\n  3:      147,       308,       215,        50;\n  4:     1029,      2891,      3045,      1425,       250;\n  5:     7203,     25382,     35770,     25200,      8875,      1250;\n  6:    50421,    213689,    377300,    355250,    188125,     53125,      6250;\n  7:   352947,   1747928,   3709545,   4373250,   3093125,   1312500,    309375,    31250;\n  8:  2470629,  14000231,  34706455,  49160475,  43518125,  24653125,   8728125,  1765625,  156250;\n  9: 17294403, 110354762, 312946340, 517655600, 550429250, 390162500, 184362500, 56000000, 9921875, 781250;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (7*n-7+(5*n-5)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((7*n - 7 + (5*n - 5)*x)*P[n - 1, x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((7*n - 7 + (5*n - 5)*x)*v[i - 1])/(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-cedb1276760d9bb1","steps":["basis A007318"],"edges":["e-3c1a0c86d879ec10"],"text":"A193726 \u2192 basis A007318"},"headline":"A193726 \u2192 basis A007318","chain":null}