{"id":"n-984f75403d58b29b","revision":0,"first_index":0,"fingerprint":"f6-4358062586543316","name":"shift_x_plus_one of T(n, k) = E(n, k)*2^k where E(n,k) are the Eulerian numbers A173018, for n > 0 and 0 <= k <= n-1, additionally T(0,0) = 1.","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":false,"rows":[["13","16","4"],["75","134","68","8"],["541","1268","984","272","16"],["4683","13586","14248","6384","1072","32"],["47293","163560","216444","135232","39216","4224","64"],["545835","2191934","3505020","2818968","1178384","234912","16704","128"],["7087261","32410396","60785392","59898448","32907040","9796672","1391872","66304","256"],["102247563","524566250","1129081088","1318351808","900147296","359905856","79301120","8216576","263936","512"],["1622632573","9228981584","22423353396","30284929920","24758721696","12485395968","3779445888","632297472","48503040","1052672","1024"],["28091567595","175430841494","474921849652","728574493224","694477837536","423828596160","164295898752","38671029504","5000402688","286797312","4203520","2048"],["526858348381","3583832435844","10697270176584","18377717534480","20023409785200","14364155969664","6812974430976","2086231681536","389129521920","39387591680","1699743744","16797696","4096"],["10641342970443","78315755164994","255521499565080","486045220364688","596084717619280","491913401183712","276539643622656","104785995947520","25847680419072","3874297781760","309816727552","10097627136","67153920","8192"],["230283190977853","1823124238332088","6455068463192620","13469081526378560","18366868041632336","17149244107221632","11162875329523392","5056140822190080","1561373665324800","314822593316864","38322087191552","2437337464832","60115578880","268533760","16384"],["5315654681981355","45045837476248814","172016945673978284","390684101456807096","586482228887991664","611481750654317152","452709275672906176","239002174830723456","88994465472463104","22735300815899136","3789382494880768","377624810039296","19194630885376","358544973824","1073954816","32768"],["130370767029135901","1177442182762911212","4823780353114456800","11846963998282795600","19416504691253223680","22365989351052071616","18570222425673505792","11210255887647100160","4898371628862097920","1522429427505996800","325508485996797952","45242254367944704","3714003123077120","151390741872640","2141603758080","4295426048","65536"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["33","285","3081","40005","606033","10491885","204343641","4422082005","105265315233","2733583519485","76902684021801","2329889536156005","75629701786875633","2618654297178083085","96336948993312237561"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["13","16","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["5","4"],["2","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1","-3","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":188464,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-eb0dc7264c9d6fc9"]},{"id":188465,"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-eb0dc7264c9d6fc9"]},{"id":188466,"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-eb0dc7264c9d6fc9"]},{"id":188467,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-eb0dc7264c9d6fc9"]}],"region_exact":{},"incoming_edges":[{"id":"e-eb0dc7264c9d6fc9","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-984f75403d58b29b","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"4dba09cfeda10c38a26e05f232532f45bea6c3b59ee9e16f8865252788146f67","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-481b4b083ef06726","role":"sequence","name":"T(n, k) = E(n, k)*2^k where E(n,k) are the Eulerian numbers A173018, for n > 0 and 0 <= k <= n-1, additionally T(0,0) = 1.","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-3df32afac9fe0cb1","kind":"fingerprint_related","node_a_id":"n-cf84c1880a723950","node_b_id":"n-984f75403d58b29b","fingerprint":"f6-4358062586543316","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4358062586543316"},"other_id":"n-cf84c1880a723950","other":{"id":"n-cf84c1880a723950","name":"shift_x_plus_one of Triangle T(n, k) = 2^(k-1) * E(n, k-1) where E(n,k) are the Eulerian numbers A173018, read by rows.","first_index":1,"fingerprint":"f6-4358062586543316","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","2"],["13","16","4"]],"card_rows":[["1"],["3","2"],["13","16","4"],["75","134","68","8"]],"row_sums_preview":["1","5","33","285","3081","40005","606033","10491885"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-423e26f84d00b286","steps":["x \u21a6 x+1"],"edges":["e-72d2df1b7f31ee08"],"text":"A142075 \u2192 x \u21a6 x+1"},"headline":"A142075 \u2192 x \u21a6 x+1","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:          13,           16,            4;\n  1:          75,          134,           68,            8;\n  2:         541,         1268,          984,          272,           16;\n  3:        4683,        13586,        14248,         6384,         1072,           32;\n  4:       47293,       163560,       216444,       135232,        39216,         4224,           64;\n  5:      545835,      2191934,      3505020,      2818968,      1178384,       234912,        16704,         128;\n  6:     7087261,     32410396,     60785392,     59898448,     32907040,      9796672,      1391872,       66304,        256;\n  7:   102247563,    524566250,   1129081088,   1318351808,    900147296,    359905856,     79301120,     8216576,     263936,       512;\n  8:  1622632573,   9228981584,  22423353396,  30284929920,  24758721696,  12485395968,   3779445888,   632297472,   48503040,   1052672,    1024;\n  9: 28091567595, 175430841494, 474921849652, 728574493224, 694477837536, 423828596160, 164295898752, 38671029504, 5000402688, 286797312, 4203520, 2048;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2*n+5+(2*n+4)*x)*P(n-1,x) - (1+3*x+2*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 13 + 16*x + 4*x^2.","mathematica":"P[0, x_] := 13 + 16*x + 4*x^2;\nP[n_, x_] := P[n, x] = Expand[(2*n + 5 + (2*n + 4)*x)*P[n - 1, x] - (1 + 3*x + 2*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 13 + 16*x + 4*x^2; for(n = 1, N, my(i = n + 1); v[i] = (2*n + 5 + (2*n + 4)*x)*v[i - 1] - (1 + 3*x + 2*x^2)*deriv(v[i - 1], x)); 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-481b4b083ef06726","steps":["x \u21a6 x+1"],"edges":["e-eb0dc7264c9d6fc9"],"text":"A156365 \u2192 x \u21a6 x+1"},"headline":"A156365 \u2192 x \u21a6 x+1","chain":null}