{"id":"n-b74f038f49f19922","revision":1,"first_index":0,"fingerprint":"f6-9b5c4299a56ccb0f","name":"shift_x_plus_one of Triangle read by rows: T(n,k), 0 <= k <= n, gives the coefficients of the Charlier polynomials (with parameter a=1), ordered by rising powers.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":false,"rows":[["1"],["0","1"],["-2","-1","1"],["4","-3","-3","1"],["-4","17","2","-6","1"],["-4","-57","24","21","-10","1"],["44","179","-189","-45","65","-15","1"],["-236","-631","1145","-66","-365","148","-21","1"],["1300","2749","-7134","1967","1969","-1281","287","-28","1"],["-8276","-15013","49516","-22276","-10500","10885","-3388","502","-36","1"],["61484","99351","-389317","230330","52534","-95655","38507","-7626","816","-45","1"],["-523804","-766883","3447845","-2447581","-179510","889349","-443457","109245","-15390","1255","-55","1"],["5024036","6719697","-34021374","27607276","-1103379","-8814489","5305292","-1553640","268743","-28655","1848","-66","1"],["-53478980","-65642849","370154200","-333490133","43181454","93108952","-66713052","22528787","-4578486","596288","-50116","2627","-78","1"],["624890236","705802299","-4401348213","4319024065","-879401729","-1043832076","886104540","-337836323","78286703","-11929060","1221924","-83343","3627","-91","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","1","-2","-1","10","-25","40","-25","-170","1415","-9620","71015","-594710","5618615","-59097440"],"nonn":false,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["0","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0"],["-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","1"],["-1","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":164890,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, +inf)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)","region_exact":null,"edges":["e-5aedd40a059ef120"]},{"id":164891,"position":1,"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-5aedd40a059ef120"]}],"region_exact":{},"incoming_edges":[{"id":"e-5aedd40a059ef120","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-b74f038f49f19922","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":"bcec7e76323ecd1333f54b3d79e35b91bc4ae42bfff4e00abe56fe30d3d15b78","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-4ae73ef2b17633d9","role":"sequence","name":"Triangle read by rows: T(n,k), 0 <= k <= n, gives the coefficients of the Charlier polynomials (with parameter a=1), ordered by rising powers.","uses":{"claim":"real_rooted","from_index":0,"region":{}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)"}]}],"used_by":[],"used_by_total":0,"links":[],"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:     1;\n  1:     0,      1;\n  2:    -2,     -1,     1;\n  3:     4,     -3,    -3,      1;\n  4:    -4,     17,     2,     -6,      1;\n  5:    -4,    -57,    24,     21,    -10,     1;\n  6:    44,    179,  -189,    -45,     65,   -15,     1;\n  7:  -236,   -631,  1145,    -66,   -365,   148,   -21,   1;\n  8:  1300,   2749, -7134,   1967,   1969, -1281,   287, -28,   1;\n  9: -8276, -15013, 49516, -22276, -10500, 10885, -3388, 502, -36, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -n*P(n-2,x) - (n-1-x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := x;\nP[n_, x_] := P[n, x] = Expand[-n*P[n - 2, x] - (n - 1 - x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = x; for(n = 2, N, my(i = n + 1); v[i] = -n*v[i - 2] - (n - 1 - 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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-4ae73ef2b17633d9","steps":["x \u21a6 x+1"],"edges":["e-5aedd40a059ef120"],"text":"A137338 \u2192 x \u21a6 x+1"},"headline":"A137338 \u2192 x \u21a6 x+1","chain":null}