{"id":"n-ff6e5b4b9776bc17","revision":0,"first_index":0,"fingerprint":"f6-2e6f63c0c32e8505","name":"basis_map_A013610 of basis_map_A013609 of Triangle read by rows: triangle of repeated stepped binomial coefficients.","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","6"],["12","42","36"],["36","198","360","216"],["144","1008","2628","3024","1296"],["432","3888","13932","24840","22032","7776"],["1728","18144","79056","182952","237168","163296","46656"],["5184","64800","346032","1023192","1809216","1912896","1119744","279936"],["20736","290304","1772928","6168960","13376016","18506880","15956352","7838208","1679616"],["62208","995328","7060608","29144448","77141808","135776736","158910336","119252736","52068096","10077696"],["248832","4354560","34214400","158941440","483433920","1005957792","1450301760","1430472960","923788800","352719360","60466176"],["746496","14556672","128770560","682110720","2403950400","5918476896","10386652032","12993229440","11354204160","6600890880","2297714688","362797056"],["2985984","62705664","602422272","3501066240","13708465920","38097609984","77057469504","114292829952","123376193280","94528788480","48796204032","15237476352","2176782336"],["8957952","206032896","2183500800","14117732352","62131795200","196543625472","459758068416","805223306880","1055885559552","1023843525120","713561342976","338489653248","97955205120","13060694016"],["35831808","877879296","9970200576","69571934208","333233574912","1158965273088","3018294026496","5979441638016","9054882079488","10430687457792","8997306522624","5635326670848","2422758739968","639974006784","78364164096"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","9","90","810","8100","72900","729000","6561000","65610000","590490000","5904900000","53144100000","531441000000","4782969000000","47829690000000"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["3","6"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["12","42","36"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":202021,"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-0ba6d2f79357fe24"]},{"id":202022,"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-0ba6d2f79357fe24"]},{"id":202023,"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-0ba6d2f79357fe24"]}],"region_exact":{},"incoming_edges":[{"id":"e-0ba6d2f79357fe24","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-ff6e5b4b9776bc17","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":"b429fc340ecd583a722e9f54106177bf7cdb0d09dd7262d3d37bd6b0f265613f","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-e42a65125928b4c5","role":"sequence","name":"basis_map_A013609 of Triangle read by rows: triangle of repeated stepped binomial coefficients.","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":[{"id":"l-6119d80307eaf8a8","kind":"fingerprint_related","node_a_id":"n-b5d38d742ea2388f","node_b_id":"n-ff6e5b4b9776bc17","fingerprint":"f6-2e6f63c0c32e8505","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2e6f63c0c32e8505"},"other_id":"n-b5d38d742ea2388f","other":{"id":"n-b5d38d742ea2388f","name":"mobius_x_over_x_plus_one of shift_x_plus_one of Triangle read by rows: triangle of repeated stepped binomial coefficients.","first_index":0,"fingerprint":"f6-2e6f63c0c32e8505","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","2"],["2","7","6"]],"card_rows":[["1"],["1","2"],["2","7","6"],["2","11","20","12"]],"row_sums_preview":["1","3","15","45","225","675","3375","10125"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-35bd700768cac55f","steps":["basis A007318","x \u21a6 x/(x+1)"],"edges":["e-01d2a4c51ed4f80c","e-0c1e92eaafcc40ce"],"text":"A103633 \u2192 basis A007318 \u2192 x \u21a6 x/(x+1)"},"headline":"A103633 \u2192 basis A007318 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-887d9e5872bd6253","kind":"fingerprint_related","node_a_id":"n-0d95f7d056b9311c","node_b_id":"n-ff6e5b4b9776bc17","fingerprint":"f6-2e6f63c0c32e8505","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2e6f63c0c32e8505"},"other_id":"n-0d95f7d056b9311c","other":{"id":"n-0d95f7d056b9311c","name":"basis_map_A013610 of Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.","first_index":0,"fingerprint":"f6-2e6f63c0c32e8505","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["6","21","18"],["18","99","180","108"],["36","252","657","756","324"]],"card_rows":[["6","21","18"],["18","99","180","108"],["36","252","657","756","324"],["108","972","3483","6210","5508","1944"]],"row_sums_preview":["45","405","2025","18225","91125","820125","4100625","36905625"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5a70a6e97131d437","steps":["basis A013610"],"edges":["e-f5362ff16cbef54d"],"text":"A064861 \u2192 basis A013610"},"headline":"A064861 \u2192 basis A013610","chain":null}}],"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,      6;\n  2:    12,     42,      36;\n  3:    36,    198,     360,      216;\n  4:   144,   1008,    2628,     3024,     1296;\n  5:   432,   3888,   13932,    24840,    22032,      7776;\n  6:  1728,  18144,   79056,   182952,   237168,    163296,     46656;\n  7:  5184,  64800,  346032,  1023192,  1809216,   1912896,   1119744,    279936;\n  8: 20736, 290304, 1772928,  6168960, 13376016,  18506880,  15956352,   7838208,  1679616;\n  9: 62208, 995328, 7060608, 29144448, 77141808, 135776736, 158910336, 119252736, 52068096, 10077696;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (12+42*x+36*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 6*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 6*x;\nP[n_, x_] := P[n, x] = Expand[(12 + 42*x + 36*x^2)*P[n - 2, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = (12 + 42*x + 36*x^2)*v[i - 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-35bd700768cac55f","steps":["basis A013609","basis A013610"],"edges":["e-69752228c1d1e91e","e-0ba6d2f79357fe24"],"text":"A103633 \u2192 basis A013609 \u2192 basis A013610"},"headline":"A103633 \u2192 basis A013609 \u2192 basis A013610","chain":null}
