{"id":"n-41513f0d6e6cd398","revision":0,"first_index":1,"fingerprint":"f6-cf9b351a67dba233","name":"basis_map_A013611 of basis_map_A007318 of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).","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"],["4","4"],["15","32","16"],["56","184","192","64"],["209","928","1488","1024","256"],["780","4364","9472","9984","5120","1024"],["2911","19648","53856","76800","60160","24576","4096"],["10864","85872","284544","512640","542720","337920","114688","16384"],["40545","367296","1427808","3111936","4161280","3497984","1806336","524288","65536"],["151316","1545492","6895872","17646336","28550144","30299136","21102592","9306112","2359296","262144"],["564719","6419936","32337648","95056896","180624640","231899136","203800576","121110528","46596096","10485760","1048576"],["2107560","26393128","148134464","491931840","1074176000","1619795968","1721696256","1290338304","668467200","228065280","46137344","4194304"],["7865521","107582816","665772720","2465208320","6083806720","10543988736","13162168320","11927027712","7788625920","3575644160","1095761920","201326592","16777216"],["29354524","435400220","2945287680","12031992320","33121884160","64891385856","93102931968","99066445824","78194147328","45229015040","18639486976","5184159744","872415232","67108864"],["109552575","1751436160","12856978880","57443911680","174531699200","381509091328","618815102976","756750483456","701253746688","490117005312","254378246144","95093260288","24209522688","3758096384","268435456"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","8","63","496","3905","30744","242047","1905632","15003009","118118440","929944511","7321437648","57641556673","453811015736","3572846569215"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["4","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["4","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":204963,"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-3d4ec2f4d87d4ff6"]},{"id":204964,"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-3d4ec2f4d87d4ff6"]},{"id":204965,"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-3d4ec2f4d87d4ff6"]},{"id":204966,"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-3d4ec2f4d87d4ff6"]}],"region_exact":{},"incoming_edges":[{"id":"e-3d4ec2f4d87d4ff6","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-41513f0d6e6cd398","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":"5ae0a67cae8965afeb308607bf097038a8d54b2cc94bac16fbe9af789e966243","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-e7f9db5cff35db04","role":"sequence","name":"basis_map_A007318 of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).","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-30c98f6697ac36e4","kind":"fingerprint_related","node_a_id":"n-1f449c58b1930aa5","node_b_id":"n-41513f0d6e6cd398","fingerprint":"f6-cf9b351a67dba233","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-cf9b351a67dba233"},"other_id":"n-1f449c58b1930aa5","other":{"id":"n-1f449c58b1930aa5","name":"basis_map_A013611 of A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).","first_index":0,"fingerprint":"f6-cf9b351a67dba233","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","4"],["15","32","16"]],"card_rows":[["1"],["4","4"],["15","32","16"],["56","184","192","64"]],"row_sums_preview":["1","8","63","496","3905","30744","242047","1905632"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f287bed89e07609f","steps":["basis A013611"],"edges":["e-362ae2b737f320ab"],"text":"A125662 \u2192 basis A013611"},"headline":"A125662 \u2192 basis A013611","chain":null}},{"id":"l-b00f15a70871ce3a","kind":"fingerprint_related","node_a_id":"n-dac117530db899a6","node_b_id":"n-41513f0d6e6cd398","fingerprint":"f6-cf9b351a67dba233","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-cf9b351a67dba233"},"other_id":"n-dac117530db899a6","other":{"id":"n-dac117530db899a6","name":"reversal of basis_map_A013611 of A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).","first_index":0,"fingerprint":"f6-cf9b351a67dba233","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","4"],["16","32","15"]],"card_rows":[["1"],["4","4"],["16","32","15"],["64","192","184","56"]],"row_sums_preview":["1","8","63","496","3905","30744","242047","1905632"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f287bed89e07609f","steps":["basis A013611","reverse"],"edges":["e-362ae2b737f320ab","e-ef60665cf0397dfb"],"text":"A125662 \u2192 basis A013611 \u2192 reverse"},"headline":"A125662 \u2192 basis A013611 \u2192 reverse","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:      4,       4;\n   3:     15,      32,      16;\n   4:     56,     184,     192,       64;\n   5:    209,     928,    1488,     1024,      256;\n   6:    780,    4364,    9472,     9984,     5120,     1024;\n   7:   2911,   19648,   53856,    76800,    60160,    24576,     4096;\n   8:  10864,   85872,  284544,   512640,   542720,   337920,   114688,   16384;\n   9:  40545,  367296, 1427808,  3111936,  4161280,  3497984,  1806336,  524288,   65536;\n  10: 151316, 1545492, 6895872, 17646336, 28550144, 30299136, 21102592, 9306112, 2359296, 262144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -P(n-2,x) + (4+4*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 4 + 4*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 4 + 4*x;\nP[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (4 + 4*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] = 4 + 4*x; for(n = 3, N, my(i = n); v[i] = -v[i - 2] + (4 + 4*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-976cdb6e4a693ec8","steps":["x \u21a6 x+1","basis A013611"],"edges":["e-05347468080deadb","e-3d4ec2f4d87d4ff6"],"text":"A078812 \u2192 x \u21a6 x+1 \u2192 basis A013611"},"headline":"A078812 \u2192 x \u21a6 x+1 \u2192 basis A013611","chain":null}