{"id":"n-20c6b6131d05ee73","revision":0,"first_index":0,"fingerprint":"f6-0b8646f5e24f04bb","name":"basis_map_A013609 of Triangle T(n,k) read by rows: coefficients (in compressed forms) in order of decreasing exponents of polynomials p_n(t) related to Hultman numbers.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["2","2"],["6","10"],["24","62","32"],["120","406","336"],["720","3096","4036","1440"],["5040","25992","44004","24352"],["40320","245232","530196","466528","129024"],["362880","2529648","6538356","7407136","3091968"],["3628800","28683360","87593976","127131592","84970176","19353600"],["39916800","351804960","1229418216","2125800872","1814611136","609822720"],["479001600","4672149120","18527833152","37901480856","41555446336","22491021312","4379443200"],["6227020800","66485957760","293711969088","686364988344","893500095872","613114863104","172779061248"],["87178291200","1013311434240","4955936003712","13150167273408","20275031605264","17910101536512","8170420368384","1394852659200"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","4","16","118","862","9292","99388","1411300","19929988","351361504","6171374704","130006375576","2732183956216","66956999171920"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1"]],"recurrence":{"denominator":{"coeffs":[["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["1","0"],["-1","0"],["-1","-2"],["1","2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1"],["2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":199387,"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-a4bd38943928866f"]},{"id":199388,"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-a4bd38943928866f"]},{"id":199389,"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-a4bd38943928866f"]},{"id":199390,"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-a4bd38943928866f"]}],"region_exact":{},"incoming_edges":[{"id":"e-a4bd38943928866f","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-20c6b6131d05ee73","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":"d116eb65f0143b4be6c42bd99cb48bc4af01245833e74c0d1a8e9b06986464ff","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-136f914693403ae6","role":"sequence","name":"Triangle T(n,k) read by rows: coefficients (in compressed forms) in order of decreasing exponents of polynomials p_n(t) related to Hultman numbers.","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;\n  2:      2,       2;\n  3:      6,      10;\n  4:     24,      62,      32;\n  5:    120,     406,     336;\n  6:    720,    3096,    4036,    1440;\n  7:   5040,   25992,   44004,   24352;\n  8:  40320,  245232,  530196,  466528,  129024;\n  9: 362880, 2529648, 6538356, 7407136, 3091968;\n","recurrence_sentence":"The row polynomials satisfy (n+2)*P(n,x) = (n^3-n^2-n+1+(2*n^3-2*n^2)*x)*P(n-2,x) + (2*n+1)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((n^3 - n^2 - n + 1 + (2*n^3 - 2*n^2)*x)*P[n - 2, x] + (2*n + 1)*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; for(n = 2, N, my(i = n + 1); v[i] = ((n^3 - n^2 - n + 1 + (2*n^3 - 2*n^2)*x)*v[i - 2] + (2*n + 1)*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-136f914693403ae6","steps":["basis A013609"],"edges":["e-a4bd38943928866f"],"text":"A185263 \u2192 basis A013609"},"headline":"A185263 \u2192 basis A013609","chain":null}