{"id":"n-790aec5277b4ba1f","revision":0,"first_index":1,"fingerprint":"f6-a264a61b871c93de","name":"basis_map_A013610 of basis_map_A013609 of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["3","6"],["12","42","36"],["45","234","396","216"],["171","1170","2952","3240","1296"],["648","5508","18468","30456","24624","7776"],["2457","24948","104328","229608","279936","178848","46656"],["9315","109998","551124","1516968","2474064","2387232","1259712","279936"],["35316","475470","2776032","9172440","18741456","24222240","19315584","8678016","1679616"],["133893","2024190","13494276","52031160","127783008","207121536","221382720","150325632","58786560","10077696"],["507627","8514234","63808884","281232648","806794992","1573178112","2110157568","1921200768","1135420416","393030144","60466176"],["1924560","35464392","295140024","1463610384","4803316848","10948366944","17676418752","20203820928","16011779328","8374565376","2600045568","362797056"],["7296561","146529000","1340718480","7388222544","27299393712","71225306208","134488999296","185094523008","184191729408","129185985024","60587108352","17051461632","2176782336"],["27663363","601306902","5999535864","36370649808","149419129248","439137282816","949538293056","1528887540096","1832400589824","1613902703616","1014924764160","431365699584","111015899136","13060694016"],["104879772","2453267250","26507778444","175318143120","792708802992","2591398904832","6314257411200","11646109943424","16333669336320","17328819981312","13685067749376","7798322718720","3030081011712","718338170880","78364164096"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","9","90","891","8829","87480","866781","8588349","85096170","843160671","8354311569","82777250160","820184055561","8126651751489","80521522263450"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["3","6"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["3","6"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3","6"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":202070,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-8057e590b282a885"]},{"id":202071,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-8057e590b282a885"]},{"id":202072,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-8057e590b282a885"]},{"id":202073,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-8057e590b282a885"]}],"region_exact":{},"incoming_edges":[{"id":"e-8057e590b282a885","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-790aec5277b4ba1f","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":"baf7c8a8d75e3b2296afb91ec9a62825f04ef890f5b405516be6e9af3d250aa7","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-273f45fc84d8c52d","role":"sequence","name":"basis_map_A013609 of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-3d499df97a2f8990","kind":"fingerprint_related","node_a_id":"n-835e923f3e04ffe3","node_b_id":"n-790aec5277b4ba1f","fingerprint":"f6-a264a61b871c93de","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a264a61b871c93de"},"other_id":"n-835e923f3e04ffe3","other":{"id":"n-835e923f3e04ffe3","name":"reversal of basis_map_A013609 of Riordan array (1, 3+3x).","first_index":0,"fingerprint":"f6-a264a61b871c93de","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["6","3"],["36","42","12"]],"card_rows":[["1"],["6","3"],["36","42","12"],["216","396","234","45"]],"row_sums_preview":["1","9","90","891","8829","87480","866781","8588349"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-016d7724d51f6416","steps":["basis A013609","reverse"],"edges":["e-2339e629d7f9e0b9","e-063f74d808b40c9f"],"text":"A099093 \u2192 basis A013609 \u2192 reverse"},"headline":"A099093 \u2192 basis A013609 \u2192 reverse","chain":null}},{"id":"l-c115148840d44366","kind":"fingerprint_related","node_a_id":"n-f8c44f01f6cd4a34","node_b_id":"n-790aec5277b4ba1f","fingerprint":"f6-a264a61b871c93de","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a264a61b871c93de"},"other_id":"n-f8c44f01f6cd4a34","other":{"id":"n-f8c44f01f6cd4a34","name":"basis_map_A013609 of Riordan array (1, 3+3x).","first_index":0,"fingerprint":"f6-a264a61b871c93de","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","6"],["12","42","36"]],"card_rows":[["1"],["3","6"],["12","42","36"],["45","234","396","216"]],"row_sums_preview":["1","9","90","891","8829","87480","866781","8588349"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-016d7724d51f6416","steps":["basis A013609"],"edges":["e-2339e629d7f9e0b9"],"text":"A099093 \u2192 basis A013609"},"headline":"A099093 \u2192 basis A013609","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:      1;\n   2:      3,       6;\n   3:     12,      42,       36;\n   4:     45,     234,      396,      216;\n   5:    171,    1170,     2952,     3240,      1296;\n   6:    648,    5508,    18468,    30456,     24624,      7776;\n   7:   2457,   24948,   104328,   229608,    279936,    178848,     46656;\n   8:   9315,  109998,   551124,  1516968,   2474064,   2387232,   1259712,    279936;\n   9:  35316,  475470,  2776032,  9172440,  18741456,  24222240,  19315584,   8678016,  1679616;\n  10: 133893, 2024190, 13494276, 52031160, 127783008, 207121536, 221382720, 150325632, 58786560, 10077696;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+6*x)*P(n-2,x) + (3+6*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + 6*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 3 + 6*x;\nP[n_, x_] := P[n, x] = Expand[(3 + 6*x)*P[n - 2, x] + (3 + 6*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + 6*x; for(n = 3, N, my(i = n); v[i] = (3 + 6*x)*v[i - 2] + (3 + 6*x)*v[i - 1]); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-392f49b496f79fbf","steps":["basis A013609","basis A013610"],"edges":["e-6b05245c845c9a6d","e-8057e590b282a885"],"text":"A026729 \u2192 basis A013609 \u2192 basis A013610"},"headline":"A026729 \u2192 basis A013609 \u2192 basis A013610","chain":null}