{"id":"n-9c7781808b7d96fc","revision":0,"first_index":0,"fingerprint":"f6-50eecff2ba79363c","name":"shift_x_plus_one of Triangle T(n, k) read by rows: T(n, k) = (Sum_{i=0..k} (-1)^i * binomial(k, i) * (n+i)^n) * (-1)^k / k! for 0 <= k <= n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2","1"],["10","7","1"],["77","64","15","1"],["799","741","223","26","1"],["10427","10451","3685","575","40","1"],["163967","174266","68716","12940","1235","57","1"],["3017562","3358587","1440698","310296","36435","2345","77","1"],["63625324","73519151","33695537","8036658","1096151","87962","4074","100","1"],["1512354975","1802767972","871840143","225425761","34486746","3250695","189546","6618","126","1"],["40012800675","48968736895","24762093031","6838296740","1146415915","122362989","8451177","374220","10200","155","1"],["1166271373797","1459943502513","766642569671","223669312691","40421004415","4766846029","375533697","19824057","689205","15070","187","1"],["37134022033885","47406716877768","25714046249068","7860324731600","1512792558926","193701914934","16969151914","1027654716","42811692","1199495","21505","222","1"],["1282405154139046","1665652920437949","929289621379704","295715643422420","60052517504122","8241098245094","788362562702","53364705394","2562236391","86381152","1991847","29809","260","1"],["47745103281852282","62968645789028635","36011430681512469","11868131428465676","2524684247407686","367654974158472","37888537306620","2813609958016","151631628577","5914055147","164655491","3179176","40313","301","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","18","157","1790","25179","421182","8166001","180064958","4450322583","121859541998","3662110671333","119839645205726","4242201333035891","161526510263248862"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"],["1","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["2","2"],["1","0"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["1","1"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":185287,"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-aebc80b0839dd915"]},{"id":185288,"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-aebc80b0839dd915"]},{"id":185289,"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-aebc80b0839dd915"]}],"region_exact":{},"incoming_edges":[{"id":"e-aebc80b0839dd915","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-9c7781808b7d96fc","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":"6c639ff831d7a95726957fcdac7d24fec446a129761fd013556fd50ec7b28841","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-735e311cbe9a718e","role":"sequence","name":"Triangle T(n, k) read by rows: T(n, k) = (Sum_{i=0..k} (-1)^i * binomial(k, i) * (n+i)^n) * (-1)^k / k! for 0 <= k <= n.","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":[],"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:          2,          1;\n  2:         10,          7,         1;\n  3:         77,         64,        15,         1;\n  4:        799,        741,       223,        26,        1;\n  5:      10427,      10451,      3685,       575,       40,       1;\n  6:     163967,     174266,     68716,     12940,     1235,      57,      1;\n  7:    3017562,    3358587,   1440698,    310296,    36435,    2345,     77,    1;\n  8:   63625324,   73519151,  33695537,   8036658,  1096151,   87962,   4074,  100,   1;\n  9: 1512354975, 1802767972, 871840143, 225425761, 34486746, 3250695, 189546, 6618, 126, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n+1+x)*P(n-1,x) + (n+2+2*x)*d/dx P(n-1,x) + (1+x)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(n + 1 + x)*P[n - 1, x] + (n + 2 + 2*x)*D[P[n - 1, x], x] + (1 + x)*D[P[n - 1, x], {x, 2}]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (n + 1 + x)*v[i - 1] + (n + 2 + 2*x)*deriv(v[i - 1], x) + (1 + x)*deriv(deriv(v[i - 1], x), 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-735e311cbe9a718e","steps":["x \u21a6 x+1"],"edges":["e-aebc80b0839dd915"],"text":"A397048 \u2192 x \u21a6 x+1"},"headline":"A397048 \u2192 x \u21a6 x+1","chain":null}
