{"id":"n-13054822f21023a5","revision":0,"first_index":0,"fingerprint":"f6-a155e4872033def2","name":"one_plus_derivative of Triangle for first differences of Catalan 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real-rootedness)","region_exact":null,"edges":["e-c09f12c6fe03a5a2"]}],"region_exact":{},"incoming_edges":[{"id":"e-c09f12c6fe03a5a2","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-13054822f21023a5","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"2650c8317358a55ef282553821eabf2f7c97e7860d404ffc96066a6d778c5f94","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-a88a492d75692c65","role":"sequence","name":"Triangle for first differences of Catalan numbers.","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:   3,    1;\n  2:   8,    7,     1;\n  3:  18,   32,    12,     1;\n  4:  35,  110,    82,    18,     1;\n  5:  61,  305,   395,   170,    25,     1;\n  6:  98,  721,  1470,  1085,   310,    33,    1;\n  7: 148, 1512,  4508,  5194,  2520,   518,   42,    1;\n  8: 213, 2892, 11928, 19992, 15036,  5208,  812,   52,  1;\n  9: 295, 5145, 28134, 65016, 70812, 37800, 9870, 1212, 63, 1;\n","recurrence_sentence":"The row polynomials satisfy (4*n^3+22*n^2+34*n+12)*P(n,x) = (44*n^3+92*n^2-851*n-884-(52*n^3+188*n^2-1210*n-1496)*x-(12*n^3-60*n^2-24*n+288)*x^2+(4*n^3+76*n^2-274*n-376)*x^3+(16*n^3-40*n^2-109*n+52)*x^4)*P(n-5,x) - (16*n^3+56*n^2-375*n-420-(32*n^3+208*n^2-1016*n-1312)*x+(288*n^2-798*n-1416)*x^2+(32*n^3-176*n^2+48*n+576)*x^3-(16*n^3-40*n^2-109*n+52)*x^4)*d/dx P(n-5,x) + (12*n^3+690*n^2-1208*n-2098+(176*n^3-156*n^2-540*n+505)*x-(4*n^3+238*n^2-344*n-609)*x^2-(40*n^3+48*n^2-268*n-99)*x^3-(16*n^3-8*n^2-64*n+35)*x^4)*P(n-4,x) + (64*n^3+176*n^2-504*n-582-(32*n^3+160*n^2-164*n-391)*x-(16*n^3-56*n^2-176*n-9)*x^2-(80*n^2-100*n-217)*x^3-(16*n^3-8*n^2-64*n+35)*x^4)*d/dx P(n-4,x) - (168*n^3+904*n^2+356*n-342+(184*n^3-348*n^2-454*n+480)*x-(44*n^3+248*n^2-31*n-63)*x^2-(36*n^3+36*n^2-72*n+30)*x^3+(3*n-3)*x^4)*P(n-3,x) - (96*n^3+192*n^2+18*n-6+(32*n^3-80*n^2-24*n+150)*x-(32*n^3+272*n^2+71*n-123)*x^2-(32*n^3+32*n^2-58*n+24)*x^3+(3*n-3)*x^4)*d/dx P(n-3,x) + (120*n^3+116*n^2+26*n+62+(48*n^3-52*n^2-216*n-137)*x-(28*n^3+70*n^2-2*n-24)*x^2+(6*n+3)*x^3)*P(n-2,x) + (64*n^3+80*n^2+48*n+54+(32*n^3-128*n^2-316*n-161)*x-(16*n^3+40*n^2-14*n-24)*x^2+(6*n+3)*x^3)*d/dx P(n-2,x) - (4*n^3-28*n^2-119*n-94-(12*n^3+48*n^2+52*n+8)*x+(3*n+6)*x^2)*P(n-1,x) - (16*n^3+8*n^2-99*n-102+(8*n+16)*x+(3*n+6)*x^2)*d/dx P(n-1,x) for n >= 5, with P(0,x) = 1, P(1,x) = 3 + x, P(2,x) = 8 + 7*x + x^2, P(3,x) = 18 + 32*x + 12*x^2 + x^3, P(4,x) = 35 + 110*x + 82*x^2 + 18*x^3 + x^4.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + x;\nP[2, x_] := 8 + 7*x + x^2;\nP[3, x_] := 18 + 32*x + 12*x^2 + x^3;\nP[4, x_] := 35 + 110*x + 82*x^2 + 18*x^3 + x^4;\nP[n_, x_] := P[n, x] = Expand[Together[((44*n^3 + 92*n^2 - 851*n - 884 - (52*n^3 + 188*n^2 - 1210*n - 1496)*x - (12*n^3 - 60*n^2 - 24*n + 288)*x^2 + (4*n^3 + 76*n^2 - 274*n - 376)*x^3 + (16*n^3 - 40*n^2 - 109*n + 52)*x^4)*P[n - 5, x] - (16*n^3 + 56*n^2 - 375*n - 420 - (32*n^3 + 208*n^2 - 1016*n - 1312)*x + (288*n^2 - 798*n - 1416)*x^2 + (32*n^3 - 176*n^2 + 48*n + 576)*x^3 - (16*n^3 - 40*n^2 - 109*n + 52)*x^4)*D[P[n - 5, x], x] + (12*n^3 + 690*n^2 - 1208*n - 2098 + (176*n^3 - 156*n^2 - 540*n + 505)*x - (4*n^3 + 238*n^2 - 344*n - 609)*x^2 - (40*n^3 + 48*n^2 - 268*n - 99)*x^3 - (16*n^3 - 8*n^2 - 64*n + 35)*x^4)*P[n - 4, x] + (64*n^3 + 176*n^2 - 504*n - 582 - (32*n^3 + 160*n^2 - 164*n - 391)*x - (16*n^3 - 56*n^2 - 176*n - 9)*x^2 - (80*n^2 - 100*n - 217)*x^3 - (16*n^3 - 8*n^2 - 64*n + 35)*x^4)*D[P[n - 4, x], x] - (168*n^3 + 904*n^2 + 356*n - 342 + (184*n^3 - 348*n^2 - 454*n + 480)*x - (44*n^3 + 248*n^2 - 31*n - 63)*x^2 - (36*n^3 + 36*n^2 - 72*n + 30)*x^3 + (3*n - 3)*x^4)*P[n - 3, x] - (96*n^3 + 192*n^2 + 18*n - 6 + (32*n^3 - 80*n^2 - 24*n + 150)*x - (32*n^3 + 272*n^2 + 71*n - 123)*x^2 - (32*n^3 + 32*n^2 - 58*n + 24)*x^3 + (3*n - 3)*x^4)*D[P[n - 3, x], x] + (120*n^3 + 116*n^2 + 26*n + 62 + (48*n^3 - 52*n^2 - 216*n - 137)*x - (28*n^3 + 70*n^2 - 2*n - 24)*x^2 + (6*n + 3)*x^3)*P[n - 2, x] + (64*n^3 + 80*n^2 + 48*n + 54 + (32*n^3 - 128*n^2 - 316*n - 161)*x - (16*n^3 + 40*n^2 - 14*n - 24)*x^2 + (6*n + 3)*x^3)*D[P[n - 2, x], x] - (4*n^3 - 28*n^2 - 119*n - 94 - (12*n^3 + 48*n^2 + 52*n + 8)*x + (3*n + 6)*x^2)*P[n - 1, x] - (16*n^3 + 8*n^2 - 99*n - 102 + (8*n + 16)*x + (3*n + 6)*x^2)*D[P[n - 1, x], x])/(4*n^3 + 22*n^2 + 34*n + 12)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + x; v[3] = 8 + 7*x + x^2; v[4] = 18 + 32*x + 12*x^2 + x^3; v[5] = 35 + 110*x + 82*x^2 + 18*x^3 + x^4; for(n = 5, N, my(i = n + 1); v[i] = ((44*n^3 + 92*n^2 - 851*n - 884 - (52*n^3 + 188*n^2 - 1210*n - 1496)*x - (12*n^3 - 60*n^2 - 24*n + 288)*x^2 + (4*n^3 + 76*n^2 - 274*n - 376)*x^3 + (16*n^3 - 40*n^2 - 109*n + 52)*x^4)*v[i - 5] - (16*n^3 + 56*n^2 - 375*n - 420 - (32*n^3 + 208*n^2 - 1016*n - 1312)*x + (288*n^2 - 798*n - 1416)*x^2 + (32*n^3 - 176*n^2 + 48*n + 576)*x^3 - (16*n^3 - 40*n^2 - 109*n + 52)*x^4)*deriv(v[i - 5], x) + (12*n^3 + 690*n^2 - 1208*n - 2098 + (176*n^3 - 156*n^2 - 540*n + 505)*x - (4*n^3 + 238*n^2 - 344*n - 609)*x^2 - (40*n^3 + 48*n^2 - 268*n - 99)*x^3 - (16*n^3 - 8*n^2 - 64*n + 35)*x^4)*v[i - 4] + (64*n^3 + 176*n^2 - 504*n - 582 - (32*n^3 + 160*n^2 - 164*n - 391)*x - (16*n^3 - 56*n^2 - 176*n - 9)*x^2 - (80*n^2 - 100*n - 217)*x^3 - (16*n^3 - 8*n^2 - 64*n + 35)*x^4)*deriv(v[i - 4], x) - (168*n^3 + 904*n^2 + 356*n - 342 + (184*n^3 - 348*n^2 - 454*n + 480)*x - (44*n^3 + 248*n^2 - 31*n - 63)*x^2 - (36*n^3 + 36*n^2 - 72*n + 30)*x^3 + (3*n - 3)*x^4)*v[i - 3] - (96*n^3 + 192*n^2 + 18*n - 6 + (32*n^3 - 80*n^2 - 24*n + 150)*x - (32*n^3 + 272*n^2 + 71*n - 123)*x^2 - (32*n^3 + 32*n^2 - 58*n + 24)*x^3 + (3*n - 3)*x^4)*deriv(v[i - 3], x) + (120*n^3 + 116*n^2 + 26*n + 62 + (48*n^3 - 52*n^2 - 216*n - 137)*x - (28*n^3 + 70*n^2 - 2*n - 24)*x^2 + (6*n + 3)*x^3)*v[i - 2] + (64*n^3 + 80*n^2 + 48*n + 54 + (32*n^3 - 128*n^2 - 316*n - 161)*x - (16*n^3 + 40*n^2 - 14*n - 24)*x^2 + (6*n + 3)*x^3)*deriv(v[i - 2], x) - (4*n^3 - 28*n^2 - 119*n - 94 - (12*n^3 + 48*n^2 + 52*n + 8)*x + (3*n + 6)*x^2)*v[i - 1] - (16*n^3 + 8*n^2 - 99*n - 102 + (8*n + 16)*x + (3*n + 6)*x^2)*deriv(v[i - 1], x))/(4*n^3 + 22*n^2 + 34*n + 12)); 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-a88a492d75692c65","steps":["D+1"],"edges":["e-c09f12c6fe03a5a2"],"text":"A119308 \u2192 D+1"},"headline":"A119308 \u2192 D+1","chain":null}