{"id":"n-f36fde97d71e8e6b","revision":1,"first_index":0,"fingerprint":"f6-8065765a9ebbaf7d","name":"gamma_polynomial of Triangle read by rows: T(n,k) (1<=k<=n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = 7.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["1","14"],["1","140"],["1","1162","1960"],["1","9352","61936"],["1","74886","1321824","867104"],["1","599172","24020976","56087360"],["1","4793474","402256680","2242802912","785223040"],["1","38347904","6436502016","71868038144","85569949696"],["1","306783358","100305624832","2031651980288","5499986143232","1197979295744"],["1","2454267004","1538944108576","53122016052224","273272109049856","197126866657280"],["1","19634136186","23393399271144","1319500875789376","11643432286101504","18573995779756032","2759776133201920"],["1","157073089656","353649768133200","31649079985734400","448497609863251200","1320696056092901376","638742255560851456"],["1","1256584717430","5328935777805024","740839630470940000","16108040524636256000","78939890709772033536","82935151345018470400","8942391577851920384"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","15","141","3123","71289","2263815","80707509","3435076107","163912837761","8830229827455","525062390134941","34320118108256163","2439938808343961289","187671643980112142775"],"nonn":true,"definitions":[{"position":0,"source":"derived:gamma_polynomial","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","-14","0"],["0","14","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","7","-28"],["0","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162654,"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-68c3c312d2db929f"]},{"id":162655,"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-68c3c312d2db929f"]},{"id":162656,"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-68c3c312d2db929f"]}],"region_exact":{},"incoming_edges":[{"id":"e-68c3c312d2db929f","operation":{"index_shift":0,"operation":"gamma_polynomial","scale":"1"},"operation_detail":"scale 1, shift 0","conclusion_id":"n-f36fde97d71e8e6b","premise_count":1,"theorem":"gamma criterion (RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff): if P = sum_j gamma_j x^j (1 + x)^(d - 2j) with deg Gamma <= d/2, then P is nonzero and real-rooted with roots in (-inf, 0] if and only if Gamma(z) = sum_j gamma_j z^j is; Gamma(c z) with c != 0 has the roots of Gamma divided by c","lean_route":null,"lean_declarations":[],"certificate_hash":"f5793560d5bf437112d0fba1d5810474cca5930098e5e48b13a6d0850670e719","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-0af3ffe51a3d7181","role":"sequence","name":"Triangle read by rows: T(n,k) (1<=k<=n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = 7.","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: 1;\n  2: 1,       14;\n  3: 1,      140;\n  4: 1,     1162,       1960;\n  5: 1,     9352,      61936;\n  6: 1,    74886,    1321824,      867104;\n  7: 1,   599172,   24020976,    56087360;\n  8: 1,  4793474,  402256680,  2242802912,   785223040;\n  9: 1, 38347904, 6436502016, 71868038144, 85569949696;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(14*n-14)*x)*P(n-1,x) + (7*x-28*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + (14*n - 14)*x)*P[n - 1, x] + (7*x - 28*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (1 + (14*n - 14)*x)*v[i - 1] + (7*x - 28*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-0af3ffe51a3d7181","steps":["\u03b3"],"edges":["e-68c3c312d2db929f"],"text":"A142462 \u2192 \u03b3"},"headline":"A142462 \u2192 \u03b3","chain":null}