{"id":"n-47ecd85a6f8f5bd0","revision":1,"first_index":0,"fingerprint":"f6-46db95fe876b6051","name":"basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j).","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"],["7","1"],["49","15","1"],["343","169","24","1"],["2401","1695","385","34","1"],["16807","15961","5160","725","45","1"],["117649","144495","62401","12410","1220","57","1"],["823543","1273609","706104","186501","25830","1904","70","1"],["5764801","11012415","7628545","2571114","470631","48678","2814","84","1"],["40353607","93864121","79669320","33339685","7748055","1054767","85260","3990","99","1"],["282475249","791266575","810888001","413066170","118568290","20405259","2163147","141120","5475","115","1"],["1977326743","6612607849","8089258584","4941549701","1717317360","363431398","48526170","4138827","223245","7315","132","1"],["13841287201","54878189535","79415935105","57504755594","23832040661","6078494136","994271608","106469748","7487502","340285","9559","150","1"],["96889010407","452866803481","769621605480","654463491045","319657202865","96773970293","19004025040","2484848080","218782278","12932062","502788","12259","169","1"],["678223072849","3719823438255","7379461252801","7314256515930","4170692722560","1480944846381","343826295813","53791898160","5766582250","425695270","21479458","723450","15470","189","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","8","65","537","4516","38699","338233","3017562","27499083","256118905","2438979402","23754387325","236659281085","2411993186248","25147234538837"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["7","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":166010,"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-845ce04339675149"]},{"id":166011,"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-845ce04339675149"]},{"id":166012,"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-845ce04339675149"]}],"region_exact":{},"incoming_edges":[{"id":"e-845ce04339675149","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-47ecd85a6f8f5bd0","premise_count":1,"theorem":"ray basis maps: for L f = (a + b t) f + kappa (t - h) f', b > 0, kappa >= 0, and g real-rooted with every root <= h, of multiplicity m at h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in (-inf, h] if and only if S lies in (-inf, a + b h + kappa m] (RealRooted rayBasisMap_maps_realRootedIn_iff; the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"3c9675c296ad0be39f364f39b701a68e225f20629d1dde162dfbdf6a78c47bea","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-22693abd431a71f2","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j).","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-8755042e0fd5bda6","kind":"fingerprint_related","node_a_id":"n-47ecd85a6f8f5bd0","node_b_id":"n-3df315a348347923","fingerprint":"f6-46db95fe876b6051","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-46db95fe876b6051"},"other_id":"n-3df315a348347923","other":{"id":"n-3df315a348347923","name":"basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.","first_index":0,"fingerprint":"f6-46db95fe876b6051","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["196","60","4"],["2744","1352","192","8"],["38416","27120","6160","544","16"]],"card_rows":[["196","60","4"],["2744","1352","192","8"],["38416","27120","6160","544","16"],["537824","510752","165120","23200","1440","32"]],"row_sums_preview":["260","4296","72256","1238368","21646912","386247936","7039765248","131132879360"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ea9b82d727693fb6","steps":["basis A008277"],"edges":["e-81c568f66e8f893e"],"text":"A038328 \u2192 basis A008277"},"headline":"A038328 \u2192 basis A008277","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:        7,        1;\n  2:       49,       15,        1;\n  3:      343,      169,       24,        1;\n  4:     2401,     1695,      385,       34,       1;\n  5:    16807,    15961,     5160,      725,      45,       1;\n  6:   117649,   144495,    62401,    12410,    1220,      57,     1;\n  7:   823543,  1273609,   706104,   186501,   25830,    1904,    70,    1;\n  8:  5764801, 11012415,  7628545,  2571114,  470631,   48678,  2814,   84,  1;\n  9: 40353607, 93864121, 79669320, 33339685, 7748055, 1054767, 85260, 3990, 99, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (7+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(7 + x)*P[n - 1, x] + x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (7 + x)*v[i - 1] + x*deriv(v[i - 1], x)); 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-22693abd431a71f2","steps":["basis A008277"],"edges":["e-845ce04339675149"],"text":"A038255 \u2192 basis A008277"},"headline":"A038255 \u2192 basis A008277","chain":null}