{"id":"n-4ef858fd88b37fab","revision":1,"first_index":0,"fingerprint":"f6-9629d1e9c04dbfbe","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["225","180","36"],["3375","4050","1620","216"],["50625","81000","48600","12960","1296"],["759375","1518750","1215000","486000","97200","7776"],["11390625","27337500","27337500","14580000","4374000","699840","46656"],["170859375","478406250","574087500","382725000","153090000","36741600","4898880","279936"],["2562890625","8201250000","11481750000","9185400000","4592700000","1469664000","293932800","33592320","1679616"],["38443359375","138396093750","221433750000","206671500000","124002900000","49601160000","13226976000","2267481600","226748160","10077696"],["576650390625","2306601562500","4151882812500","4428675000000","3100072500000","1488034800000","496011600000","113374080000","17006112000","1511654400","60466176"],["8649755859375","38058925781250","76117851562500","91341421875000","73073137500000","40920957000000","16368382800000","4676680800000","935336160000","124711488000","9976919040","362797056"],["129746337890625","622782421875000","1370121328125000","1826828437500000","1644145593750000","1052253180000000","491051484000000","168360508800000","42090127200000","7482689280000","897922713600","65303470080","2176782336"],["1946195068359375","10120214355468750","24288514453125000","35623154531250000","35623154531250000","25648671262500000","13679291340000000","5471716536000000","1641514960800000","364781102400000","58364976384000","6367088332800","424472555520","13060694016"],["29192926025390625","163480385742187500","425049002929687500","680078404687500000","748086245156250000","598468996125000000","359081397675000000","164151496080000000","57453023628000000","15320806300800000","3064161260160000","445696183296000","44569618329600","2742745743360","78364164096"],["437893890380859375","2627363342285156250","7356617358398437500","12751470087890625000","15301764105468750000","13465552412812500000","8977034941875000000","4616760827250000000","1846704330900000000","574530236280000000","137887256707200000","25070410310400000","3342721374720000","308558896128000","17631936921600","470184984576"],["6568408355712890625","42037813476562500000","126113440429687500000","235411755468750000000","306035282109375000000","293793870825000000000","215448838605000000000","123113622060000000000","55401129927000000000","19698179529600000000","5515490268288000000","1203379694899200000","200563282483200000","24684711690240000","2115832430592000","112844396298240","2821109907456"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["441","9261","194481","4084101","85766121","1801088541","37822859361","794280046581","16679880978201","350277500542221","7355827511386641","154472377739119461","3243919932521508681","68122318582951682301","1430568690241985328321"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["225","180","36"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["15","6"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":166869,"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-f3e6c91e6b1bdc2d"]},{"id":166870,"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-f3e6c91e6b1bdc2d"]},{"id":166871,"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-f3e6c91e6b1bdc2d"]}],"region_exact":{},"incoming_edges":[{"id":"e-f3e6c91e6b1bdc2d","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-4ef858fd88b37fab","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":"1c44baa775899f4bedcc06473904d814a53c8841a2039960d4b6f594bb790e94","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-615df6ea54d3bbef","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^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-338e037a2d605f9f","kind":"fingerprint_related","node_a_id":"n-4ef858fd88b37fab","node_b_id":"n-815b996b7a3cb30d","fingerprint":"f6-9629d1e9c04dbfbe","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9629d1e9c04dbfbe"},"other_id":"n-815b996b7a3cb30d","other":{"id":"n-815b996b7a3cb30d","name":"basis_map_A013611 of Triangle of coefficients in expansion of (3+5x)^n.","first_index":0,"fingerprint":"f6-9629d1e9c04dbfbe","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","20"],["64","320","400"]],"card_rows":[["1"],["8","20"],["64","320","400"],["512","3840","9600","8000"]],"row_sums_preview":["1","28","784","21952","614656","17210368","481890304","13492928512"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c44f7114310405ad","steps":["basis A013611"],"edges":["e-96f2019a633e1196"],"text":"A013622 \u2192 basis A013611"},"headline":"A013622 \u2192 basis A013611","chain":null}},{"id":"l-35d41a340f09597a","kind":"fingerprint_related","node_a_id":"n-64a6f5553b7b7b7f","node_b_id":"n-4ef858fd88b37fab","fingerprint":"f6-9629d1e9c04dbfbe","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9629d1e9c04dbfbe"},"other_id":"n-64a6f5553b7b7b7f","other":{"id":"n-64a6f5553b7b7b7f","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.","first_index":0,"fingerprint":"f6-9629d1e9c04dbfbe","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["400","320","64"],["8000","9600","3840","512"],["160000","256000","153600","40960","4096"]],"card_rows":[["400","320","64"],["8000","9600","3840","512"],["160000","256000","153600","40960","4096"],["3200000","6400000","5120000","2048000","409600","32768"]],"row_sums_preview":["784","21952","614656","17210368","481890304","13492928512","377801998336","10578455953408"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6874f381e9c7e0a2","steps":["basis A007318"],"edges":["e-ec87a016e71e40fc"],"text":"A038334 \u2192 basis A007318"},"headline":"A038334 \u2192 basis A007318","chain":null}},{"id":"l-5cfee447cd9ca0aa","kind":"fingerprint_related","node_a_id":"n-4ef858fd88b37fab","node_b_id":"n-8afa7b2308002172","fingerprint":"f6-9629d1e9c04dbfbe","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9629d1e9c04dbfbe"},"other_id":"n-8afa7b2308002172","other":{"id":"n-8afa7b2308002172","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*1^j.","first_index":0,"fingerprint":"f6-9629d1e9c04dbfbe","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["10","4"],["100","80","16"]],"card_rows":[["1"],["10","4"],["100","80","16"],["1000","1200","480","64"]],"row_sums_preview":["1","14","196","2744","38416","537824","7529536","105413504"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3dd184aba99388f7","steps":["basis A013611"],"edges":["e-8c5f302cdd82df95"],"text":"A038291 \u2192 basis A013611"},"headline":"A038291 \u2192 basis A013611","chain":null}},{"id":"l-7e4e92099682338a","kind":"fingerprint_related","node_a_id":"n-4ef858fd88b37fab","node_b_id":"n-31eaa59be7a07b63","fingerprint":"f6-9629d1e9c04dbfbe","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9629d1e9c04dbfbe"},"other_id":"n-31eaa59be7a07b63","other":{"id":"n-31eaa59be7a07b63","name":"basis_map_A013610 of Triangle of coefficients in expansion of (1+5x)^n.","first_index":0,"fingerprint":"f6-9629d1e9c04dbfbe","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["36","180","225"],["216","1620","4050","3375"],["1296","12960","48600","81000","50625"]],"card_rows":[["36","180","225"],["216","1620","4050","3375"],["1296","12960","48600","81000","50625"],["7776","97200","486000","1215000","1518750","759375"]],"row_sums_preview":["441","9261","194481","4084101","85766121","1801088541","37822859361","794280046581"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-bbfc5a54ff13a785","steps":["basis A013610"],"edges":["e-46de8f4df4fd1cb3"],"text":"A013612 \u2192 basis A013610"},"headline":"A013612 \u2192 basis A013610","chain":null}},{"id":"l-a50733091dbe5d59","kind":"fingerprint_related","node_a_id":"n-f1a275eb676f791b","node_b_id":"n-4ef858fd88b37fab","fingerprint":"f6-9629d1e9c04dbfbe","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9629d1e9c04dbfbe"},"other_id":"n-f1a275eb676f791b","other":{"id":"n-f1a275eb676f791b","name":"mobius_x_over_x_plus_one of Triangle of coefficients in expansion of (2+3x)^n.","first_index":0,"fingerprint":"f6-9629d1e9c04dbfbe","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","5"],["4","20","25"]],"card_rows":[["1"],["2","5"],["4","20","25"],["8","60","150","125"]],"row_sums_preview":["1","7","49","343","2401","16807","117649","823543"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5733d79f45a14084","steps":["x \u21a6 x/(x+1)"],"edges":["e-867523535b04b58e"],"text":"A013620 \u2192 x \u21a6 x/(x+1)"},"headline":"A013620 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-f1f4c48ba5776714","kind":"fingerprint_related","node_a_id":"n-3fcb2ab6ab8d081a","node_b_id":"n-4ef858fd88b37fab","fingerprint":"f6-9629d1e9c04dbfbe","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9629d1e9c04dbfbe"},"other_id":"n-3fcb2ab6ab8d081a","other":{"id":"n-3fcb2ab6ab8d081a","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.","first_index":0,"fingerprint":"f6-9629d1e9c04dbfbe","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["5","2"],["25","20","4"]],"card_rows":[["1"],["5","2"],["25","20","4"],["125","150","60","8"]],"row_sums_preview":["1","7","49","343","2401","16807","117649","823543"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-01cea33b6485d9cc","steps":["basis A007318"],"edges":["e-7f6faf86ff30c5bb"],"text":"A038220 \u2192 basis A007318"},"headline":"A038220 \u2192 basis A007318","chain":null}},{"id":"l-5948e5a4d155c645","kind":"oeis_match","node_a_id":"n-4ef858fd88b37fab","node_b_id":null,"fingerprint":null,"oeis_id":"A009965","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A009965","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A009965","relation":"match_row_sums"}],"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:           225,            180,             36;\n  1:          3375,           4050,           1620,            216;\n  2:         50625,          81000,          48600,          12960,           1296;\n  3:        759375,        1518750,        1215000,         486000,          97200,           7776;\n  4:      11390625,       27337500,       27337500,       14580000,        4374000,         699840,          46656;\n  5:     170859375,      478406250,      574087500,      382725000,      153090000,       36741600,        4898880,        279936;\n  6:    2562890625,     8201250000,    11481750000,     9185400000,     4592700000,     1469664000,      293932800,      33592320,      1679616;\n  7:   38443359375,   138396093750,   221433750000,   206671500000,   124002900000,    49601160000,    13226976000,    2267481600,    226748160,     10077696;\n  8:  576650390625,  2306601562500,  4151882812500,  4428675000000,  3100072500000,  1488034800000,   496011600000,  113374080000,  17006112000,   1511654400,   60466176;\n  9: 8649755859375, 38058925781250, 76117851562500, 91341421875000, 73073137500000, 40920957000000, 16368382800000, 4676680800000, 935336160000, 124711488000, 9976919040, 362797056;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (15+6*x)*P(n-1,x) for n >= 1, with P(0,x) = 225 + 180*x + 36*x^2.","mathematica":"P[0, x_] := 225 + 180*x + 36*x^2;\nP[n_, x_] := P[n, x] = Expand[(15 + 6*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 225 + 180*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = (15 + 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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-615df6ea54d3bbef","steps":["basis A013609"],"edges":["e-f3e6c91e6b1bdc2d"],"text":"A038329 \u2192 basis A013609"},"headline":"A038329 \u2192 basis A013609","chain":null}
