{"id":"n-5d25926d30ecf6f4","revision":0,"first_index":1,"fingerprint":"f6-4a3147452b8a179e","name":"basis_map_A013609 of basis_map_A013609 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"],["5","4"],["24","40","16"],["115","292","240","64"],["551","1880","2352","1280","256"],["2640","11312","19040","15744","6400","1024"],["12649","65240","138096","153600","94720","30720","4096"],["60605","365484","932400","1304640","1081600","531456","143360","16384"],["290376","2004600","5985840","10099200","10531840","6952960","2838528","655360","65536"],["1391275","10819020","37015200","73134720","91974400","76360704","41861120","14614528","2949120","262144"],["6665999","57655600","222366240","503635200","741879040","742748160","511909888","239861760","73138176","13107200","1048576"],["31938720","304122976","1305438400","3334506240","5631961600","6604896256","5488680960","3232333824","1322188800","357826560","57671680","4194304"],["153027601","1590714160","7521317664","21390649600","40755953920","54809579520","53351079936","37876531200","19467141120","7064780800","1718616064","251658240","16777216"],["733199285","8261558228","42664006560","133704012416","283710406400","430466817024","480505036800","399554641920","247519641600","112834641920","36794531840","8128561152","1090519040","67108864"],["3512968824","42649874120","238844948048","817785438720","1912612127744","3232366131200","4071041372160","3881916825600","2816349634560","1547186995200","633592610816","187569274880","37950062592","4697620480","268435456"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","9","80","711","6319","56160","499121","4435929","39424240","350382231","3114015839","27675760320","245967827041","2186034683049","19428344320400"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["5","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["5","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":199683,"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-5c5486bf4056cc4e"]},{"id":199684,"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-5c5486bf4056cc4e"]},{"id":199685,"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-5c5486bf4056cc4e"]},{"id":199686,"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-5c5486bf4056cc4e"]}],"region_exact":{},"incoming_edges":[{"id":"e-5c5486bf4056cc4e","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-5d25926d30ecf6f4","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":"336869c2f8430de14048a5afcc6065d41455f9de95c1253459b737618754c51e","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-8fd7b2e367875124","role":"sequence","name":"basis_map_A013609 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-2ad86250f07791e0","kind":"fingerprint_related","node_a_id":"n-beaf9b3575bb564b","node_b_id":"n-5d25926d30ecf6f4","fingerprint":"f6-4a3147452b8a179e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4a3147452b8a179e"},"other_id":"n-beaf9b3575bb564b","other":{"id":"n-beaf9b3575bb564b","name":"reversal of basis_map_A013611 of Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x in increasing order).","first_index":0,"fingerprint":"f6-4a3147452b8a179e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","5"],["16","40","24"]],"card_rows":[["1"],["4","5"],["16","40","24"],["64","240","292","115"]],"row_sums_preview":["1","9","80","711","6319","56160","499121","4435929"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-30cb3cad216ed189","steps":["basis A013611","reverse"],"edges":["e-21c167d0f859d82c","e-158ade57ddd23980"],"text":"A207823 \u2192 basis A013611 \u2192 reverse"},"headline":"A207823 \u2192 basis A013611 \u2192 reverse","chain":null}},{"id":"l-4c10c380b5abd62a","kind":"fingerprint_related","node_a_id":"n-f7cf3d554e6e9f71","node_b_id":"n-5d25926d30ecf6f4","fingerprint":"f6-4a3147452b8a179e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4a3147452b8a179e"},"other_id":"n-f7cf3d554e6e9f71","other":{"id":"n-f7cf3d554e6e9f71","name":"basis_map_A013611 of Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x in increasing order).","first_index":0,"fingerprint":"f6-4a3147452b8a179e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["5","4"],["24","40","16"]],"card_rows":[["1"],["5","4"],["24","40","16"],["115","292","240","64"]],"row_sums_preview":["1","9","80","711","6319","56160","499121","4435929"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-30cb3cad216ed189","steps":["basis A013611"],"edges":["e-21c167d0f859d82c"],"text":"A207823 \u2192 basis A013611"},"headline":"A207823 \u2192 basis A013611","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:       5,        4;\n   3:      24,       40,       16;\n   4:     115,      292,      240,       64;\n   5:     551,     1880,     2352,     1280,      256;\n   6:    2640,    11312,    19040,    15744,     6400,     1024;\n   7:   12649,    65240,   138096,   153600,    94720,    30720,     4096;\n   8:   60605,   365484,   932400,  1304640,  1081600,   531456,   143360,    16384;\n   9:  290376,  2004600,  5985840, 10099200, 10531840,  6952960,  2838528,   655360,   65536;\n  10: 1391275, 10819020, 37015200, 73134720, 91974400, 76360704, 41861120, 14614528, 2949120, 262144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -P(n-2,x) + (5+4*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 5 + 4*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 5 + 4*x;\nP[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (5 + 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] = 5 + 4*x; for(n = 3, N, my(i = n); v[i] = -v[i - 2] + (5 + 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":["basis A013609","basis A013609"],"edges":["e-567d6f79aa6a87e5","e-5c5486bf4056cc4e"],"text":"A078812 \u2192 basis A013609 \u2192 basis A013609"},"headline":"A078812 \u2192 basis A013609 \u2192 basis A013609","chain":null}
