{"id":"n-37a972deb39d81b2","revision":0,"first_index":0,"fingerprint":"f6-c1efd17c58b17f52","name":"basis_map_A013610 of basis_map_A013609 of Square the entries of Pascal's 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"],["4","6"],["22","60","36"],["136","540","648","216"],["886","4632","8424","6048","1296"],["5944","38580","94320","105840","51840","7776"],["40636","315144","968760","1486080","1166400","419904","46656"],["281488","2539320","9407664","18340560","19958400","11757312","3265920","279936"],["1968934","20255280","87836112","207918144","290213280","239936256","111041280","24634368","1679616"],["13875544","160320708","796581216","2219247072","3785854464","4047454656","2657152512","997691904","181398528","10077696"],["98365972","1261170600","7064155800","22645267200","45690220800","60050471040","51350060160","27612887040","8616430080","1310100480","60466176"],["700701808","9871771272","61541924400","223147386000","520041772800","811979562240","858107416320","604989285120","272702453760","72055526400","9311791104","362797056"],["5011371964","76953306960","528453055152","2138466052800","5654338034400","10232993295360","12899417287680","11293872353280","6714466513920","2582913484800","586642839552","65303470080","2176782336"],["35961808432","597799180680","4483731990816","20031908431392","59260703673600","122023331876160","178861031331840","187488479692800","139038343741440","70931460188160","23622683774976","4669198110720","452770725888","13060694016"],["258805997752","4630214604624","37660872111408","184122782678016","602627296182912","1391548757639040","2326791238611840","2844521604587520","2534250507909120","1619289433681920","718814402482176","209708307873792","36448043433984","3108445175808","78364164096"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","10","118","1540","21286","304300","4443580","65830600","985483270","14869654300","225759595348","3444812388280","52781007848284","811510465220920","12513859077134008"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["4","6"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["4","24","36"],["-4","-24","-36"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-4","-6"],["8","12"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":202850,"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-3f1e288d7152f1e2"]},{"id":202851,"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-3f1e288d7152f1e2"]},{"id":202852,"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-3f1e288d7152f1e2"]},{"id":202853,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-3f1e288d7152f1e2"]}],"region_exact":{},"incoming_edges":[{"id":"e-3f1e288d7152f1e2","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-37a972deb39d81b2","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":"9b893dc869b693418b35df29b03be128482fc7d16bfe23058ee31aa6c525000a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-fe801e460afa1e3b","role":"sequence","name":"basis_map_A013609 of Square the entries of Pascal's 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:        4,         6;\n  2:       22,        60,        36;\n  3:      136,       540,       648,        216;\n  4:      886,      4632,      8424,       6048,       1296;\n  5:     5944,     38580,     94320,     105840,      51840,       7776;\n  6:    40636,    315144,    968760,    1486080,    1166400,     419904,      46656;\n  7:   281488,   2539320,   9407664,   18340560,   19958400,   11757312,    3265920,    279936;\n  8:  1968934,  20255280,  87836112,  207918144,  290213280,  239936256,  111041280,  24634368,   1679616;\n  9: 13875544, 160320708, 796581216, 2219247072, 3785854464, 4047454656, 2657152512, 997691904, 181398528, 10077696;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = -(4*n-4+(24*n-24)*x+(36*n-36)*x^2)*P(n-2,x) + (8*n-4+(12*n-6)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 6*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + 6*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 4 + (24*n - 24)*x + (36*n - 36)*x^2)*P[n - 2, x] + (8*n - 4 + (12*n - 6)*x)*P[n - 1, x])/(n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n - 4 + (24*n - 24)*x + (36*n - 36)*x^2)*v[i - 2] + (8*n - 4 + (12*n - 6)*x)*v[i - 1])/(n)); 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-b13531ad79e7c629","steps":["basis A013609","basis A013610"],"edges":["e-4579edc5f7a4d216","e-3f1e288d7152f1e2"],"text":"A008459 \u2192 basis A013609 \u2192 basis A013610"},"headline":"A008459 \u2192 basis A013609 \u2192 basis A013610","chain":null}
