{"id":"n-d1f210a2bc5f4d66","revision":0,"first_index":1,"fingerprint":"f6-a562d7fe2342b9e0","name":"shift_x_plus_one of Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th diagonal of the Sheffer triangle A154537 (S2[2,1] generalized Stirling2), for n >= 0.","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"],["3","2"],["29","40","12"],["471","994","644","120"],["10729","30544","30808","12672","1680"],["314427","1127330","1548872","1002992","297264","30240"],["11265701","48721656","84835620","75358336","35435376","8121984","665280"],["477107439","2416417090","5097635340","5766438120","3739152080","1367381088","253638720","17297280"],["23316504721","135351795872","335476755120","461392513280","382452665440","193654774272","57552009984","8917770240","518918400"],["1291502224563","8453441839298","24074677795984","39006263624160","39424112588320","25597235800256","10567383142656","2631825980928","348734718720","17643225600"],["79955783163629","582559940345288","1874408048152892","3496892860405504","4173045728699680","3311665543422208","1757533112044416","609639659003904","130162254467328","15017365923840","670442572800"],["5471210121890631","43915857433393186","157541289603354772","332619865660074328","457875542787444320","429848894551022912","279392364732209792","124849550446258944","37205907905938176","6930207483701760","706219011671040","28158588057600"],["410048406225261049","3595091634395158512","14226806423976395016","33536631598360962688","52323521942734337904","56750864357096759808","43708108972595517184","23962229517697523712","9215782432916764416","2400235955207958528","395504504544430080","36014915284992000","1295295050649600"],["33404608947510526827","317621040112919744354","1374450982928925490392","3578291562083238825936","6240905819593150535888","7681841132272745928672","6844707071879299515648","4452794627554158892544","2104156723123838391552","708342168465975983616","163446976763856377856","24096666386766090240","1979778391757107200","64764752532480000"],["2939051077157786881781","30122809668125308061848","141500440097266302555732","403236898795697749279616","777536178539328484829968","1071046806085833390549888","1084273414693128190981184","817031123115141474033664","458713068581143291014912","189980792904583640549376","56735833040010714627072","11727949106869760065536","1561601676125151866880","116705324545354137600","3497296636753920000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","81","2229","86433","4321125","264403953","19135067157","1598633707329","151412820940485","16031550738201489","1876385068325017461","240562135956086710497","33526103923256675890725","5046526170001593353355057"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-13","-14"],["8","8"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-2","-6","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191185,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-15baac26930f8bb6"]},{"id":191186,"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-15baac26930f8bb6"]},{"id":191187,"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-15baac26930f8bb6"]},{"id":191188,"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-15baac26930f8bb6"]}],"region_exact":{},"incoming_edges":[{"id":"e-15baac26930f8bb6","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-d1f210a2bc5f4d66","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":"9745435df87827b21c7a38b08aa17f34dd04037d423f5dfb2ba7da8add32843a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-945ed98594b7abe2","role":"sequence","name":"Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th diagonal of the Sheffer triangle A154537 (S2[2,1] generalized Stirling2), for n >= 0.","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":[],"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:             3,             2;\n   3:            29,            40,             12;\n   4:           471,           994,            644,            120;\n   5:         10729,         30544,          30808,          12672,           1680;\n   6:        314427,       1127330,        1548872,        1002992,         297264,          30240;\n   7:      11265701,      48721656,       84835620,       75358336,       35435376,        8121984,         665280;\n   8:     477107439,    2416417090,     5097635340,     5766438120,     3739152080,     1367381088,      253638720,      17297280;\n   9:   23316504721,  135351795872,   335476755120,   461392513280,   382452665440,   193654774272,    57552009984,    8917770240,    518918400;\n  10: 1291502224563, 8453441839298, 24074677795984, 39006263624160, 39424112588320, 25597235800256, 10567383142656, 2631825980928, 348734718720, 17643225600;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (8*n-13+(8*n-14)*x)*P(n-1,x) - (2+6*x+4*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(8*n - 13 + (8*n - 14)*x)*P[n - 1, x] - (2 + 6*x + 4*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (8*n - 13 + (8*n - 14)*x)*v[i - 1] - (2 + 6*x + 4*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-945ed98594b7abe2","steps":["x \u21a6 x+1"],"edges":["e-15baac26930f8bb6"],"text":"A290315 \u2192 x \u21a6 x+1"},"headline":"A290315 \u2192 x \u21a6 x+1","chain":null}
