{"id":"n-c82f794c6974cbe9","revision":0,"first_index":0,"fingerprint":"f6-6dec8cea3983a758","name":"basis_map_A013611 of derivative of Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other words, the unsigned Stirling numbers of the first kind in reverse order).","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"],["7","16"],["46","232","288"],["326","2632","7008","6144"],["2556","28832","121008","223744","153600"],["22212","324512","1885008","5438464","7787520","4423680"],["212976","3839968","28709568","113881344","252646400","297050112","144506880"],["2239344","48190176","442704576","2249873664","6829071360","12375072768","12390432768","5284823040"],["25659360","643134528","7029315264","43748556288","169535028480","418766610432","643657383936","562626035712","214035333120"],["318540960","9127228608","115910539584","856130933248","4052313127680","12744114081792","26622673043456","35613223944192","27672560271360","9512681472000"],["4261576320","137558347776","1993323563136","17073656187904","95714958860800","366891502123008","973669932720128","1766098817515520","2094925308493824","1467032902041600","460413783244800"],["61148511360","2197369847808","35817407360640","349530238483456","2268713836464640","10282771157981184","33203633244250112","76372035222634496","122603365156257792","130800777512878080","83444716409978880","24108939922636800"],["937030429440","37122623050752","672838551177984","7376763582439424","54481459211929600","285526539607130112","1088672315670360064","3042452983060889600","6184211172711727104","8914960871986298880","8650030503476330496","5071116344355717120","1358136948975206400"],["15275952518400","661786753858560","13210878603932928","160890065557880832","1333595338949703680","7944052153148620800","34984485322721234944","115311822948816388096","284437705523511951360","518519774937413058560","678912047266500968448","604504972286798856192","328019779125541601280","81906105230504755200"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","1","23","566","16110","529740","19881396","840847248","39622407696","2060067057120","117199053182880","7243956004674816","483472558627285248","34657676800842686976","2656049118140775329280"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["0","0","-16"],["0","8","32"],["-1","-8","-16"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-1","0"],["0","-8"],["2","8"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":205638,"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-57d8e2a54bff89cb"]},{"id":205639,"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-57d8e2a54bff89cb"]},{"id":205640,"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-57d8e2a54bff89cb"]},{"id":205641,"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-57d8e2a54bff89cb"]}],"region_exact":{},"incoming_edges":[{"id":"e-57d8e2a54bff89cb","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-c82f794c6974cbe9","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":"8f45316210ea4a272e79245542dc51432ce4d21bd9b5d2e40ebb199d8f3e9fc7","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-89353a17ee1d740a","role":"sequence","name":"derivative of Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other words, the unsigned Stirling numbers of the first kind in reverse order).","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-da8931fd723c35d2","kind":"fingerprint_related","node_a_id":"n-c65a1a3dd755a936","node_b_id":"n-c82f794c6974cbe9","fingerprint":"f6-6dec8cea3983a758","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6dec8cea3983a758"},"other_id":"n-c65a1a3dd755a936","other":{"id":"n-c65a1a3dd755a936","name":"derivative of basis_map_A013611 of Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other words, the unsigned Stirling numbers of the first kind in reverse order).","first_index":0,"fingerprint":"f6-6dec8cea3983a758","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["4"],["28","64"]],"card_rows":[["4"],["28","64"],["184","928","1152"],["1304","10528","28032","24576"]],"row_sums_preview":["0","4","92","2264","64440","2118960","79525584","3363388992"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4d60c3dd8627c01d","steps":["basis A013611","D"],"edges":["e-60ae4909e1eb1035","e-f3d1e3ab7b9d4cd8"],"text":"A094638 \u2192 basis A013611 \u2192 D"},"headline":"A094638 \u2192 basis A013611 \u2192 D","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:        0;\n  1:        1;\n  2:        7,        16;\n  3:       46,       232,        288;\n  4:      326,      2632,       7008,        6144;\n  5:     2556,     28832,     121008,      223744,       153600;\n  6:    22212,    324512,    1885008,     5438464,      7787520,      4423680;\n  7:   212976,   3839968,   28709568,   113881344,    252646400,    297050112,    144506880;\n  8:  2239344,  48190176,  442704576,  2249873664,   6829071360,  12375072768,  12390432768,   5284823040;\n  9: 25659360, 643134528, 7029315264, 43748556288, 169535028480, 418766610432, 643657383936, 562626035712, 214035333120;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = -(n^3+(8*n^3-8*n^2)*x+(16*n^3-32*n^2+16*n)*x^2)*P(n-2,x) + (2*n^2-1+(8*n^2-8*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + (8*n^3 - 8*n^2)*x + (16*n^3 - 32*n^2 + 16*n)*x^2)*P[n - 2, x] + (2*n^2 - 1 + (8*n^2 - 8*n)*x)*P[n - 1, x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + (8*n^3 - 8*n^2)*x + (16*n^3 - 32*n^2 + 16*n)*x^2)*v[i - 2] + (2*n^2 - 1 + (8*n^2 - 8*n)*x)*v[i - 1])/(n - 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-4d60c3dd8627c01d","steps":["D","basis A013611"],"edges":["e-74e2b15799b20e69","e-57d8e2a54bff89cb"],"text":"A094638 \u2192 D \u2192 basis A013611"},"headline":"A094638 \u2192 D \u2192 basis A013611","chain":null}
