{"id":"n-eee12ab308ecad8c","revision":0,"first_index":0,"fingerprint":"f6-4b06920c162cf6ed","name":"basis_map_A013610 of Elements in the 5-Pascal triangle (by row).","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","3"],["7","21","9"],["14","63","81","27"],["28","168","351","297","81"],["56","420","1206","1647","1053","243"],["112","1008","3672","6912","7047","3645","729"],["224","2352","10368","24840","34830","28431","12393","2187"],["448","5376","27792","80784","144180","161352","110079","41553","6561"],["896","12096","71712","244944","530712","755244","704214","413343","137781","19683"],["1792","26880","179712","705024","1796256","3102624","3674160","2939328","1515591","452709","59049"],["3584","59136","440064","1949184","5707584","11594016","16656192","16901136","11849166","5452191","1476225","177147"],["7168","129024","1057536","5218560","17262720","40310784","68094432","83770848","74401740","46451880","19309023","4782969","531441"],["14336","279552","2502144","13609728","50181120","132409728","257121216","371824992","400116024","316108980","177973686","67493007","15411789","1594323"],["28672","602112","5842944","34725888","141191424","415362816","911471616","1515013632","1915707024","1832566032","1304274312","668907072","233302599","49424013","4782969"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","37","185","925","4625","23125","115625","578125","2890625","14453125","72265625","361328125","1806640625","9033203125"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","3"],["7","21","9"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["-4","-6"],["2","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":201494,"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-2238e50cd553c2c2"]},{"id":201495,"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-2238e50cd553c2c2"]},{"id":201496,"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-2238e50cd553c2c2"]},{"id":201497,"position":3,"claim":"interlacing","from_index":3,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":3,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 3","region_exact":null,"edges":["e-2238e50cd553c2c2"]}],"region_exact":{},"incoming_edges":[{"id":"e-2238e50cd553c2c2","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-eee12ab308ecad8c","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":"2aa5d9c06593e49c6e32afec4570a125ed04b1ee847e7eba1659df798cb14dff","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":3,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-10c99b3e7ebd1948","role":"sequence","name":"Elements in the 5-Pascal triangle (by row).","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,     3;\n  2:   7,    21,     9;\n  3:  14,    63,    81,     27;\n  4:  28,   168,   351,    297,     81;\n  5:  56,   420,  1206,   1647,   1053,    243;\n  6: 112,  1008,  3672,   6912,   7047,   3645,    729;\n  7: 224,  2352, 10368,  24840,  34830,  28431,  12393,   2187;\n  8: 448,  5376, 27792,  80784, 144180, 161352, 110079,  41553,   6561;\n  9: 896, 12096, 71712, 244944, 530712, 755244, 704214, 413343, 137781, 19683;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = (2*n-4+(3*n-6)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2 + 3*x, P(2,x) = 7 + 21*x + 9*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 3*x;\nP[2, x_] := 7 + 21*x + 9*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n - 4 + (3*n - 6)*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] = 2 + 3*x; v[3] = 7 + 21*x + 9*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((2*n - 4 + (3*n - 6)*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-10c99b3e7ebd1948","steps":["basis A013610"],"edges":["e-2238e50cd553c2c2"],"text":"A028313 \u2192 basis A013610"},"headline":"A028313 \u2192 basis A013610","chain":null}